id	sid	tid	token	lemma	pos
ejpam-5974	1	1	european	european	PROPN
ejpam-5974	1	2	journal	journal	PROPN
ejpam-5974	1	3	of	of	ADP
ejpam-5974	1	4	pure	pure	ADJ
ejpam-5974	1	5	and	and	CCONJ
ejpam-5974	1	6	applied	applied	ADJ
ejpam-5974	1	7	mathematics	mathematic	NOUN
ejpam-5974	1	8	2025	2025	NUM
ejpam-5974	1	9	,	,	PUNCT
ejpam-5974	1	10	vol	vol	NOUN
ejpam-5974	1	11	.	.	PROPN
ejpam-5974	1	12	18	18	NUM
ejpam-5974	1	13	,	,	PUNCT
ejpam-5974	1	14	issue	issue	NOUN
ejpam-5974	1	15	3	3	NUM
ejpam-5974	1	16	,	,	PUNCT
ejpam-5974	1	17	article	article	NOUN
ejpam-5974	1	18	number	number	NOUN
ejpam-5974	1	19	5974	5974	NUM
ejpam-5974	1	20	issn	issn	PROPN
ejpam-5974	1	21	1307	1307	NUM
ejpam-5974	1	22	-	-	SYM
ejpam-5974	1	23	5543	5543	NUM
ejpam-5974	1	24	–	–	PUNCT
ejpam-5974	2	1	ejpam.com	ejpam.com	X
ejpam-5974	2	2	published	publish	VERB
ejpam-5974	2	3	by	by	ADP
ejpam-5974	2	4	new	new	PROPN
ejpam-5974	2	5	york	york	PROPN
ejpam-5974	2	6	business	business	PROPN
ejpam-5974	2	7	global	global	PROPN
ejpam-5974	2	8	on	on	ADP
ejpam-5974	2	9	density	density	NOUN
ejpam-5974	2	10	of	of	ADP
ejpam-5974	2	11	some	some	DET
ejpam-5974	2	12	graphs	graph	NOUN
ejpam-5974	2	13	and	and	CCONJ
ejpam-5974	2	14	their	their	PRON
ejpam-5974	2	15	mycielski	mycielski	ADJ
ejpam-5974	2	16	graph	graph	NOUN
ejpam-5974	2	17	racma	racma	PROPN
ejpam-5974	2	18	l.	l.	PROPN
ejpam-5974	2	19	sango1,∗	sango1,∗	PROPN
ejpam-5974	2	20	,	,	PUNCT
ejpam-5974	2	21	isagani	isagani	PROPN
ejpam-5974	2	22	s.	s.	PROPN
ejpam-5974	2	23	cabahug	cabahug	PROPN
ejpam-5974	2	24	,	,	PUNCT
ejpam-5974	2	25	jr.1	jr.1	PROPN
ejpam-5974	2	26	1	1	NUM
ejpam-5974	2	27	department	department	NOUN
ejpam-5974	2	28	of	of	ADP
ejpam-5974	2	29	mathematics	mathematic	NOUN
ejpam-5974	2	30	,	,	PUNCT
ejpam-5974	2	31	college	college	NOUN
ejpam-5974	2	32	of	of	ADP
ejpam-5974	2	33	arts	art	NOUN
ejpam-5974	2	34	and	and	CCONJ
ejpam-5974	2	35	sciences	science	NOUN
ejpam-5974	2	36	,	,	PUNCT
ejpam-5974	2	37	central	central	ADJ
ejpam-5974	2	38	mindanao	mindanao	PROPN
ejpam-5974	2	39	university	university	PROPN
ejpam-5974	2	40	,	,	PUNCT
ejpam-5974	2	41	university	university	NOUN
ejpam-5974	2	42	town	town	NOUN
ejpam-5974	2	43	,	,	PUNCT
ejpam-5974	2	44	musuan	musuan	PROPN
ejpam-5974	2	45	,	,	PUNCT
ejpam-5974	2	46	8710	8710	NUM
ejpam-5974	2	47	maramag	maramag	NOUN
ejpam-5974	2	48	,	,	PUNCT
ejpam-5974	2	49	bukidnon	bukidnon	NOUN
ejpam-5974	2	50	,	,	PUNCT
ejpam-5974	2	51	philippines	philippine	NOUN
ejpam-5974	2	52	abstract	abstract	ADJ
ejpam-5974	2	53	.	.	PUNCT
ejpam-5974	3	1	graph	graph	NOUN
ejpam-5974	3	2	density	density	NOUN
ejpam-5974	3	3	offers	offer	VERB
ejpam-5974	3	4	an	an	DET
ejpam-5974	3	5	immediate	immediate	ADJ
ejpam-5974	3	6	measure	measure	NOUN
ejpam-5974	3	7	of	of	ADP
ejpam-5974	3	8	how	how	SCONJ
ejpam-5974	3	9	tightly	tightly	ADV
ejpam-5974	3	10	a	a	DET
ejpam-5974	3	11	network	network	NOUN
ejpam-5974	3	12	is	be	AUX
ejpam-5974	3	13	connected	connect	VERB
ejpam-5974	3	14	.	.	PUNCT
ejpam-5974	4	1	this	this	DET
ejpam-5974	4	2	paper	paper	NOUN
ejpam-5974	4	3	gives	give	VERB
ejpam-5974	4	4	explicit	explicit	ADJ
ejpam-5974	4	5	closed	closed	ADJ
ejpam-5974	4	6	-	-	PUNCT
ejpam-5974	4	7	form	form	NOUN
ejpam-5974	4	8	formulas	formula	NOUN
ejpam-5974	4	9	for	for	ADP
ejpam-5974	4	10	the	the	DET
ejpam-5974	4	11	densities	density	NOUN
ejpam-5974	4	12	of	of	ADP
ejpam-5974	4	13	nine	nine	NUM
ejpam-5974	4	14	well	well	ADV
ejpam-5974	4	15	-	-	PUNCT
ejpam-5974	4	16	known	know	VERB
ejpam-5974	4	17	undirected	undirected	ADJ
ejpam-5974	4	18	graph	graph	NOUN
ejpam-5974	4	19	,	,	PUNCT
ejpam-5974	4	20	simple	simple	ADJ
ejpam-5974	4	21	,	,	PUNCT
ejpam-5974	4	22	and	and	CCONJ
ejpam-5974	4	23	connected	connected	ADJ
ejpam-5974	4	24	graph	graph	NOUN
ejpam-5974	4	25	families	family	NOUN
ejpam-5974	4	26	:	:	PUNCT
ejpam-5974	4	27	barbell	barbell	ADJ
ejpam-5974	4	28	,	,	PUNCT
ejpam-5974	4	29	friendship	friendship	NOUN
ejpam-5974	4	30	,	,	PUNCT
ejpam-5974	4	31	sunlet	sunlet	NOUN
ejpam-5974	4	32	,	,	PUNCT
ejpam-5974	4	33	banana	banana	NOUN
ejpam-5974	4	34	,	,	PUNCT
ejpam-5974	4	35	lollipop	lollipop	NOUN
ejpam-5974	4	36	,	,	PUNCT
ejpam-5974	4	37	gear	gear	NOUN
ejpam-5974	4	38	,	,	PUNCT
ejpam-5974	4	39	tadpole	tadpole	NOUN
ejpam-5974	4	40	,	,	PUNCT
ejpam-5974	4	41	wheel	wheel	NOUN
ejpam-5974	4	42	,	,	PUNCT
ejpam-5974	4	43	and	and	CCONJ
ejpam-5974	4	44	fan	fan	NOUN
ejpam-5974	4	45	graphs	graph	NOUN
ejpam-5974	4	46	and	and	CCONJ
ejpam-5974	4	47	their	their	PRON
ejpam-5974	4	48	corresponding	correspond	VERB
ejpam-5974	4	49	mycielski	mycielski	ADJ
ejpam-5974	4	50	graph	graph	NOUN
ejpam-5974	4	51	.	.	PUNCT
ejpam-5974	5	1	the	the	DET
ejpam-5974	5	2	result	result	NOUN
ejpam-5974	5	3	supply	supply	NOUN
ejpam-5974	5	4	ready	ready	ADJ
ejpam-5974	5	5	to	to	PART
ejpam-5974	5	6	use	use	VERB
ejpam-5974	5	7	expressions	expression	NOUN
ejpam-5974	5	8	that	that	PRON
ejpam-5974	5	9	remove	remove	VERB
ejpam-5974	5	10	the	the	DET
ejpam-5974	5	11	need	need	NOUN
ejpam-5974	5	12	to	to	PART
ejpam-5974	5	13	build	build	VERB
ejpam-5974	5	14	the	the	DET
ejpam-5974	5	15	graphs	graph	NOUN
ejpam-5974	5	16	just	just	ADV
ejpam-5974	5	17	to	to	PART
ejpam-5974	5	18	calculate	calculate	VERB
ejpam-5974	5	19	its	its	PRON
ejpam-5974	5	20	density	density	NOUN
ejpam-5974	5	21	.	.	PUNCT
ejpam-5974	6	1	all	all	DET
ejpam-5974	6	2	proofs	proof	NOUN
ejpam-5974	6	3	rely	rely	VERB
ejpam-5974	6	4	only	only	ADV
ejpam-5974	6	5	on	on	ADP
ejpam-5974	6	6	counting	count	VERB
ejpam-5974	6	7	vertices	vertex	NOUN
ejpam-5974	6	8	and	and	CCONJ
ejpam-5974	6	9	edges	edge	NOUN
ejpam-5974	6	10	and	and	CCONJ
ejpam-5974	6	11	substituting	substitute	VERB
ejpam-5974	6	12	them	they	PRON
ejpam-5974	6	13	into	into	ADP
ejpam-5974	6	14	the	the	DET
ejpam-5974	6	15	definition	definition	NOUN
ejpam-5974	6	16	of	of	ADP
ejpam-5974	6	17	density	density	NOUN
ejpam-5974	6	18	.	.	PUNCT
ejpam-5974	7	1	2020	2020	NUM
ejpam-5974	7	2	mathematics	mathematic	NOUN
ejpam-5974	7	3	subject	subject	NOUN
ejpam-5974	7	4	classifications	classification	NOUN
ejpam-5974	7	5	:	:	PUNCT
ejpam-5974	7	6	:	:	PUNCT
ejpam-5974	7	7	05c07	05c07	NOUN
ejpam-5974	7	8	,	,	PUNCT
ejpam-5974	7	9	05c12	05c12	NOUN
ejpam-5974	7	10	key	key	ADJ
ejpam-5974	7	11	words	word	NOUN
ejpam-5974	7	12	and	and	CCONJ
ejpam-5974	7	13	phrases	phrase	NOUN
ejpam-5974	7	14	:	:	PUNCT
ejpam-5974	7	15	graph	graph	NOUN
ejpam-5974	7	16	density	density	NOUN
ejpam-5974	7	17	,	,	PUNCT
ejpam-5974	7	18	mycielski	mycielski	ADJ
ejpam-5974	7	19	graph	graph	NOUN
ejpam-5974	7	20	,	,	PUNCT
ejpam-5974	7	21	special	special	ADJ
ejpam-5974	7	22	graph	graph	NOUN
ejpam-5974	7	23	families	family	NOUN
ejpam-5974	7	24	1	1	NUM
ejpam-5974	7	25	.	.	PUNCT
ejpam-5974	8	1	introduction	introduction	NOUN
ejpam-5974	8	2	graphs	graph	NOUN
ejpam-5974	8	3	model	model	NOUN
ejpam-5974	8	4	pairwise	pairwise	NOUN
ejpam-5974	8	5	relationship	relationship	NOUN
ejpam-5974	8	6	in	in	ADP
ejpam-5974	8	7	diverse	diverse	ADJ
ejpam-5974	8	8	area	area	NOUN
ejpam-5974	8	9	such	such	ADJ
ejpam-5974	8	10	as	as	ADP
ejpam-5974	8	11	sociology	sociology	NOUN
ejpam-5974	8	12	,	,	PUNCT
ejpam-5974	8	13	biology	biology	NOUN
ejpam-5974	8	14	,	,	PUNCT
ejpam-5974	8	15	and	and	CCONJ
ejpam-5974	8	16	computer	computer	NOUN
ejpam-5974	8	17	science	science	NOUN
ejpam-5974	8	18	.	.	PUNCT
ejpam-5974	9	1	a	a	DET
ejpam-5974	9	2	fundamental	fundamental	ADJ
ejpam-5974	9	3	characteristic	characteristic	NOUN
ejpam-5974	9	4	of	of	ADP
ejpam-5974	9	5	any	any	DET
ejpam-5974	9	6	graph	graph	NOUN
ejpam-5974	9	7	is	be	AUX
ejpam-5974	9	8	its	its	PRON
ejpam-5974	9	9	density	density	NOUN
ejpam-5974	9	10	which	which	PRON
ejpam-5974	9	11	ranges	range	VERB
ejpam-5974	9	12	from	from	ADP
ejpam-5974	9	13	0	0	NUM
ejpam-5974	9	14	for	for	ADP
ejpam-5974	9	15	an	an	DET
ejpam-5974	9	16	empty	empty	ADJ
ejpam-5974	9	17	graph	graph	NOUN
ejpam-5974	9	18	to	to	ADP
ejpam-5974	9	19	1	1	NUM
ejpam-5974	9	20	for	for	ADP
ejpam-5974	9	21	a	a	DET
ejpam-5974	9	22	complete	complete	ADJ
ejpam-5974	9	23	graph	graph	NOUN
ejpam-5974	9	24	.	.	PUNCT
ejpam-5974	10	1	density	density	NOUN
ejpam-5974	10	2	influences	influence	VERB
ejpam-5974	10	3	algorithmic	algorithmic	ADJ
ejpam-5974	10	4	complexity	complexity	NOUN
ejpam-5974	10	5	,	,	PUNCT
ejpam-5974	10	6	highlights	highlight	VERB
ejpam-5974	10	7	communication	communication	NOUN
ejpam-5974	10	8	bottlenecks	bottleneck	NOUN
ejpam-5974	10	9	,	,	PUNCT
ejpam-5974	10	10	and	and	CCONJ
ejpam-5974	10	11	guides	guide	VERB
ejpam-5974	10	12	community	community	NOUN
ejpam-5974	10	13	-	-	PUNCT
ejpam-5974	10	14	detection	detection	NOUN
ejpam-5974	10	15	heuristics	heuristic	NOUN
ejpam-5974	10	16	.	.	PUNCT
ejpam-5974	11	1	for	for	ADP
ejpam-5974	11	2	example	example	NOUN
ejpam-5974	11	3	,	,	PUNCT
ejpam-5974	11	4	in	in	ADP
ejpam-5974	11	5	social	social	ADJ
ejpam-5974	11	6	networks	network	NOUN
ejpam-5974	11	7	,	,	PUNCT
ejpam-5974	11	8	density	density	NOUN
ejpam-5974	11	9	measures	measure	VERB
ejpam-5974	11	10	the	the	DET
ejpam-5974	11	11	degree	degree	NOUN
ejpam-5974	11	12	of	of	ADP
ejpam-5974	11	13	interaction	interaction	NOUN
ejpam-5974	11	14	or	or	CCONJ
ejpam-5974	11	15	connection	connection	NOUN
ejpam-5974	11	16	within	within	ADP
ejpam-5974	11	17	a	a	DET
ejpam-5974	11	18	community	community	NOUN
ejpam-5974	11	19	or	or	CCONJ
ejpam-5974	11	20	group	group	NOUN
ejpam-5974	11	21	,	,	PUNCT
ejpam-5974	11	22	indicating	indicate	VERB
ejpam-5974	11	23	whether	whether	SCONJ
ejpam-5974	11	24	relationships	relationship	NOUN
ejpam-5974	11	25	are	be	AUX
ejpam-5974	11	26	sparse	sparse	ADJ
ejpam-5974	11	27	or	or	CCONJ
ejpam-5974	11	28	highly	highly	ADV
ejpam-5974	11	29	interconnected	interconnected	ADJ
ejpam-5974	11	30	[	[	X
ejpam-5974	11	31	1	1	NUM
ejpam-5974	11	32	]	]	PUNCT
ejpam-5974	11	33	.	.	PUNCT
ejpam-5974	12	1	the	the	DET
ejpam-5974	12	2	concept	concept	NOUN
ejpam-5974	12	3	of	of	ADP
ejpam-5974	12	4	graph	graph	NOUN
ejpam-5974	12	5	density	density	NOUN
ejpam-5974	12	6	serves	serve	VERB
ejpam-5974	12	7	as	as	ADP
ejpam-5974	12	8	a	a	DET
ejpam-5974	12	9	fundamental	fundamental	ADJ
ejpam-5974	12	10	metric	metric	NOUN
ejpam-5974	12	11	in	in	ADP
ejpam-5974	12	12	network	network	NOUN
ejpam-5974	12	13	analysis	analysis	NOUN
ejpam-5974	12	14	.	.	PUNCT
ejpam-5974	13	1	a	a	DET
ejpam-5974	13	2	substantial	substantial	ADJ
ejpam-5974	13	3	body	body	NOUN
ejpam-5974	13	4	of	of	ADP
ejpam-5974	13	5	research	research	NOUN
ejpam-5974	13	6	underscores	underscore	VERB
ejpam-5974	13	7	its	its	PRON
ejpam-5974	13	8	significance	significance	NOUN
ejpam-5974	13	9	in	in	ADP
ejpam-5974	13	10	both	both	CCONJ
ejpam-5974	13	11	theoretical	theoretical	ADJ
ejpam-5974	13	12	and	and	CCONJ
ejpam-5974	13	13	applied	applied	ADJ
ejpam-5974	13	14	contexts	contexts	NOUN
ejpam-5974	13	15	,	,	PUNCT
ejpam-5974	13	16	ranging	range	VERB
ejpam-5974	13	17	from	from	ADP
ejpam-5974	13	18	sociology	sociology	NOUN
ejpam-5974	13	19	,	,	PUNCT
ejpam-5974	13	20	computer	computer	NOUN
ejpam-5974	13	21	science	science	NOUN
ejpam-5974	13	22	,	,	PUNCT
ejpam-5974	13	23	and	and	CCONJ
ejpam-5974	13	24	data	datum	NOUN
ejpam-5974	13	25	mining	mining	NOUN
ejpam-5974	13	26	to	to	ADP
ejpam-5974	13	27	bioinformatics	bioinformatics	NOUN
ejpam-5974	13	28	and	and	CCONJ
ejpam-5974	13	29	communication	communication	NOUN
ejpam-5974	13	30	systems	system	NOUN
ejpam-5974	13	31	.	.	PUNCT
ejpam-5974	14	1	a	a	DET
ejpam-5974	14	2	dense	dense	ADJ
ejpam-5974	14	3	graph	graph	NOUN
ejpam-5974	14	4	often	often	ADV
ejpam-5974	14	5	suggests	suggest	VERB
ejpam-5974	14	6	a	a	DET
ejpam-5974	14	7	highly	highly	ADV
ejpam-5974	14	8	collaborative	collaborative	ADJ
ejpam-5974	14	9	or	or	CCONJ
ejpam-5974	14	10	interactive	interactive	ADJ
ejpam-5974	14	11	environment	environment	NOUN
ejpam-5974	14	12	,	,	PUNCT
ejpam-5974	14	13	whereas	whereas	SCONJ
ejpam-5974	14	14	a	a	DET
ejpam-5974	14	15	sparse	sparse	ADJ
ejpam-5974	14	16	graph	graph	NOUN
ejpam-5974	14	17	may	may	AUX
ejpam-5974	14	18	indicate	indicate	VERB
ejpam-5974	14	19	isolated	isolated	ADJ
ejpam-5974	14	20	vertices	vertex	NOUN
ejpam-5974	14	21	or	or	CCONJ
ejpam-5974	14	22	limited	limited	ADJ
ejpam-5974	14	23	connectivity	connectivity	NOUN
ejpam-5974	14	24	.	.	PUNCT
ejpam-5974	15	1	a	a	DET
ejpam-5974	15	2	newly	newly	ADV
ejpam-5974	15	3	introduced	introduce	VERB
ejpam-5974	15	4	parameter	parameter	NOUN
ejpam-5974	15	5	,	,	PUNCT
ejpam-5974	15	6	referred	refer	VERB
ejpam-5974	15	7	to	to	ADP
ejpam-5974	15	8	in	in	ADP
ejpam-5974	15	9	[	[	X
ejpam-5974	15	10	2	2	NUM
ejpam-5974	15	11	]	]	PUNCT
ejpam-5974	15	12	,	,	PUNCT
ejpam-5974	15	13	has	have	AUX
ejpam-5974	15	14	recently	recently	ADV
ejpam-5974	15	15	been	be	AUX
ejpam-5974	15	16	utilized	utilize	VERB
ejpam-5974	15	17	to	to	PART
ejpam-5974	15	18	study	study	VERB
ejpam-5974	15	19	the	the	DET
ejpam-5974	15	20	density	density	NOUN
ejpam-5974	15	21	of	of	ADP
ejpam-5974	15	22	various	various	ADJ
ejpam-5974	15	23	graphs	graph	NOUN
ejpam-5974	15	24	,	,	PUNCT
ejpam-5974	15	25	including	include	VERB
ejpam-5974	15	26	corona	corona	NOUN
ejpam-5974	15	27	graphs	graph	NOUN
ejpam-5974	15	28	.	.	PUNCT
ejpam-5974	16	1	in	in	ADP
ejpam-5974	16	2	addition	addition	NOUN
ejpam-5974	16	3	to	to	ADP
ejpam-5974	16	4	density	density	NOUN
ejpam-5974	16	5	,	,	PUNCT
ejpam-5974	16	6	centrality	centrality	NOUN
ejpam-5974	16	7	measures	measure	NOUN
ejpam-5974	16	8	are	be	AUX
ejpam-5974	16	9	key	key	ADJ
ejpam-5974	16	10	tools	tool	NOUN
ejpam-5974	16	11	used	use	VERB
ejpam-5974	16	12	to	to	PART
ejpam-5974	16	13	analyze	analyze	VERB
ejpam-5974	16	14	the	the	DET
ejpam-5974	16	15	relative	relative	ADJ
ejpam-5974	16	16	importance	importance	NOUN
ejpam-5974	16	17	of	of	ADP
ejpam-5974	16	18	nodes	node	NOUN
ejpam-5974	16	19	within	within	ADP
ejpam-5974	16	20	a	a	DET
ejpam-5974	16	21	graph	graph	NOUN
ejpam-5974	16	22	.	.	PUNCT
ejpam-5974	17	1	∗corresponding	∗corresponde	VERB
ejpam-5974	17	2	author	author	NOUN
ejpam-5974	17	3	.	.	PUNCT
ejpam-5974	18	1	doi	doi	NOUN
ejpam-5974	18	2	:	:	PUNCT
ejpam-5974	18	3	https://doi.org/10.29020/nybg.ejpam.v18i3.5974	https://doi.org/10.29020/nybg.ejpam.v18i3.5974	ADJ
ejpam-5974	18	4	email	email	NOUN
ejpam-5974	18	5	addresses	address	NOUN
ejpam-5974	18	6	:	:	PUNCT
ejpam-5974	18	7	racmasango@gmail.com	racmasango@gmail.com	X
ejpam-5974	18	8	(	(	PUNCT
ejpam-5974	18	9	r.	r.	PROPN
ejpam-5974	18	10	sango	sango	PROPN
ejpam-5974	18	11	)	)	PUNCT
ejpam-5974	18	12	,	,	PUNCT
ejpam-5974	18	13	isaganicabahugjr@cmu.edu.ph	isaganicabahugjr@cmu.edu.ph	PROPN
ejpam-5974	18	14	(	(	PUNCT
ejpam-5974	18	15	i.	i.	PROPN
ejpam-5974	18	16	cabahug	cabahug	PROPN
ejpam-5974	18	17	,	,	PUNCT
ejpam-5974	18	18	jr	jr	PROPN
ejpam-5974	18	19	.	.	PUNCT
ejpam-5974	18	20	)	)	PUNCT
ejpam-5974	18	21	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5974	19	1	1	1	NUM
ejpam-5974	19	2	copyright	copyright	NOUN
ejpam-5974	19	3	:	:	PUNCT
ejpam-5974	19	4	©	©	PROPN
ejpam-5974	19	5	2025	2025	NUM
ejpam-5974	19	6	the	the	DET
ejpam-5974	19	7	author(s	author(s	NOUN
ejpam-5974	19	8	)	)	PUNCT
ejpam-5974	19	9	.	.	PUNCT
ejpam-5974	20	1	(	(	PUNCT
ejpam-5974	20	2	cc	cc	NOUN
ejpam-5974	20	3	by	by	ADP
ejpam-5974	20	4	-	-	PUNCT
ejpam-5974	20	5	nc	nc	PROPN
ejpam-5974	20	6	4.0	4.0	NUM
ejpam-5974	20	7	)	)	PUNCT
ejpam-5974	20	8	r.	r.	NOUN
ejpam-5974	20	9	sango	sango	PROPN
ejpam-5974	20	10	,	,	PUNCT
ejpam-5974	20	11	i.	i.	NOUN
ejpam-5974	20	12	cabahug	cabahug	PROPN
ejpam-5974	20	13	,	,	PUNCT
ejpam-5974	20	14	jr	jr	PROPN
ejpam-5974	20	15	/	/	SYM
ejpam-5974	20	16	eur	eur	PROPN
ejpam-5974	20	17	.	.	PUNCT
ejpam-5974	21	1	j.	j.	PROPN
ejpam-5974	21	2	pure	pure	PROPN
ejpam-5974	21	3	appl	appl	PROPN
ejpam-5974	21	4	.	.	PROPN
ejpam-5974	21	5	math	math	PROPN
ejpam-5974	21	6	,	,	PUNCT
ejpam-5974	21	7	18	18	NUM
ejpam-5974	21	8	(	(	PUNCT
ejpam-5974	21	9	3	3	NUM
ejpam-5974	21	10	)	)	PUNCT
ejpam-5974	21	11	(	(	PUNCT
ejpam-5974	21	12	2025	2025	NUM
ejpam-5974	21	13	)	)	PUNCT
ejpam-5974	21	14	,	,	PUNCT
ejpam-5974	21	15	5974	5974	NUM
ejpam-5974	21	16	2	2	NUM
ejpam-5974	21	17	of	of	ADP
ejpam-5974	21	18	16	16	NUM
ejpam-5974	21	19	these	these	PRON
ejpam-5974	21	20	include	include	VERB
ejpam-5974	21	21	degree	degree	NOUN
ejpam-5974	21	22	centrality	centrality	NOUN
ejpam-5974	21	23	,	,	PUNCT
ejpam-5974	21	24	which	which	PRON
ejpam-5974	21	25	counts	count	VERB
ejpam-5974	21	26	the	the	DET
ejpam-5974	21	27	number	number	NOUN
ejpam-5974	21	28	of	of	ADP
ejpam-5974	21	29	direct	direct	ADJ
ejpam-5974	21	30	connections	connection	NOUN
ejpam-5974	21	31	a	a	DET
ejpam-5974	21	32	node	node	NOUN
ejpam-5974	21	33	has	have	VERB
ejpam-5974	21	34	;	;	PUNCT
ejpam-5974	21	35	betweenness	betweenness	NOUN
ejpam-5974	21	36	centrality	centrality	NOUN
ejpam-5974	21	37	,	,	PUNCT
ejpam-5974	21	38	which	which	PRON
ejpam-5974	21	39	quantifies	quantify	VERB
ejpam-5974	21	40	how	how	SCONJ
ejpam-5974	21	41	often	often	ADV
ejpam-5974	21	42	a	a	DET
ejpam-5974	21	43	node	node	NOUN
ejpam-5974	21	44	appears	appear	VERB
ejpam-5974	21	45	on	on	ADP
ejpam-5974	21	46	the	the	DET
ejpam-5974	21	47	shortest	short	ADJ
ejpam-5974	21	48	paths	path	NOUN
ejpam-5974	21	49	between	between	ADP
ejpam-5974	21	50	other	other	ADJ
ejpam-5974	21	51	nodes	node	NOUN
ejpam-5974	21	52	;	;	PUNCT
ejpam-5974	21	53	closeness	closeness	NOUN
ejpam-5974	21	54	centrality	centrality	NOUN
ejpam-5974	21	55	,	,	PUNCT
ejpam-5974	21	56	which	which	PRON
ejpam-5974	21	57	assesses	assess	VERB
ejpam-5974	21	58	a	a	DET
ejpam-5974	21	59	node	node	NOUN
ejpam-5974	21	60	’s	’s	PART
ejpam-5974	21	61	average	average	ADJ
ejpam-5974	21	62	distance	distance	NOUN
ejpam-5974	21	63	to	to	ADP
ejpam-5974	21	64	all	all	DET
ejpam-5974	21	65	other	other	ADJ
ejpam-5974	21	66	nodes	node	NOUN
ejpam-5974	21	67	in	in	ADP
ejpam-5974	21	68	the	the	DET
ejpam-5974	21	69	network	network	NOUN
ejpam-5974	21	70	;	;	PUNCT
ejpam-5974	21	71	and	and	CCONJ
ejpam-5974	21	72	eigenvector	eigenvector	NOUN
ejpam-5974	21	73	centrality	centrality	NOUN
ejpam-5974	21	74	,	,	PUNCT
ejpam-5974	21	75	which	which	PRON
ejpam-5974	21	76	considers	consider	VERB
ejpam-5974	21	77	not	not	PART
ejpam-5974	21	78	just	just	ADV
ejpam-5974	21	79	the	the	DET
ejpam-5974	21	80	number	number	NOUN
ejpam-5974	21	81	of	of	ADP
ejpam-5974	21	82	a	a	DET
ejpam-5974	21	83	node	node	NOUN
ejpam-5974	21	84	’s	’s	PART
ejpam-5974	21	85	connections	connection	NOUN
ejpam-5974	21	86	,	,	PUNCT
ejpam-5974	21	87	but	but	CCONJ
ejpam-5974	21	88	also	also	ADV
ejpam-5974	21	89	the	the	DET
ejpam-5974	21	90	importance	importance	NOUN
ejpam-5974	21	91	of	of	ADP
ejpam-5974	21	92	the	the	DET
ejpam-5974	21	93	nodes	node	NOUN
ejpam-5974	21	94	it	it	PRON
ejpam-5974	21	95	is	be	AUX
ejpam-5974	21	96	connected	connect	VERB
ejpam-5974	21	97	to	to	ADP
ejpam-5974	21	98	.	.	PUNCT
ejpam-5974	22	1	the	the	DET
ejpam-5974	22	2	concept	concept	NOUN
ejpam-5974	22	3	of	of	ADP
ejpam-5974	22	4	centralities	centrality	NOUN
ejpam-5974	22	5	and	and	CCONJ
ejpam-5974	22	6	measures	measure	NOUN
ejpam-5974	22	7	has	have	AUX
ejpam-5974	22	8	also	also	ADV
ejpam-5974	22	9	been	be	AUX
ejpam-5974	22	10	examined	examine	VERB
ejpam-5974	22	11	in	in	ADP
ejpam-5974	22	12	other	other	ADJ
ejpam-5974	22	13	works	work	NOUN
ejpam-5974	22	14	,	,	PUNCT
ejpam-5974	22	15	such	such	ADJ
ejpam-5974	22	16	as	as	ADP
ejpam-5974	22	17	[	[	X
ejpam-5974	22	18	3	3	NUM
ejpam-5974	22	19	]	]	PUNCT
ejpam-5974	22	20	,	,	PUNCT
ejpam-5974	22	21	[	[	X
ejpam-5974	22	22	4	4	X
ejpam-5974	22	23	]	]	PUNCT
ejpam-5974	22	24	and	and	CCONJ
ejpam-5974	22	25	[	[	X
ejpam-5974	22	26	5	5	NUM
ejpam-5974	22	27	]	]	PUNCT
ejpam-5974	22	28	.	.	PUNCT
ejpam-5974	23	1	these	these	DET
ejpam-5974	23	2	metrics	metric	NOUN
ejpam-5974	23	3	,	,	PUNCT
ejpam-5974	23	4	when	when	SCONJ
ejpam-5974	23	5	used	use	VERB
ejpam-5974	23	6	in	in	ADP
ejpam-5974	23	7	conjunction	conjunction	NOUN
ejpam-5974	23	8	with	with	ADP
ejpam-5974	23	9	graph	graph	NOUN
ejpam-5974	23	10	density	density	NOUN
ejpam-5974	23	11	,	,	PUNCT
ejpam-5974	23	12	provide	provide	VERB
ejpam-5974	23	13	a	a	DET
ejpam-5974	23	14	more	more	ADV
ejpam-5974	23	15	comprehensive	comprehensive	ADJ
ejpam-5974	23	16	understanding	understanding	NOUN
ejpam-5974	23	17	of	of	ADP
ejpam-5974	23	18	network	network	NOUN
ejpam-5974	23	19	structure	structure	NOUN
ejpam-5974	23	20	,	,	PUNCT
ejpam-5974	23	21	influence	influence	NOUN
ejpam-5974	23	22	,	,	PUNCT
ejpam-5974	23	23	and	and	CCONJ
ejpam-5974	23	24	flow	flow	NOUN
ejpam-5974	23	25	dynamics	dynamic	NOUN
ejpam-5974	23	26	.	.	PUNCT
ejpam-5974	24	1	moreover	moreover	ADV
ejpam-5974	24	2	,	,	PUNCT
ejpam-5974	24	3	graph	graph	NOUN
ejpam-5974	24	4	density	density	NOUN
ejpam-5974	24	5	plays	play	VERB
ejpam-5974	24	6	a	a	DET
ejpam-5974	24	7	central	central	ADJ
ejpam-5974	24	8	role	role	NOUN
ejpam-5974	24	9	in	in	ADP
ejpam-5974	24	10	the	the	DET
ejpam-5974	24	11	study	study	NOUN
ejpam-5974	24	12	of	of	ADP
ejpam-5974	24	13	large	large	ADJ
ejpam-5974	24	14	-	-	PUNCT
ejpam-5974	24	15	scale	scale	NOUN
ejpam-5974	24	16	and	and	CCONJ
ejpam-5974	24	17	complex	complex	ADJ
ejpam-5974	24	18	networks	network	NOUN
ejpam-5974	24	19	.	.	PUNCT
ejpam-5974	25	1	in	in	ADP
ejpam-5974	25	2	biological	biological	ADJ
ejpam-5974	25	3	systems	system	NOUN
ejpam-5974	25	4	,	,	PUNCT
ejpam-5974	25	5	for	for	ADP
ejpam-5974	25	6	instance	instance	NOUN
ejpam-5974	25	7	,	,	PUNCT
ejpam-5974	25	8	analyzing	analyze	VERB
ejpam-5974	25	9	protein	protein	NOUN
ejpam-5974	25	10	-	-	PUNCT
ejpam-5974	25	11	protein	protein	NOUN
ejpam-5974	25	12	interaction	interaction	NOUN
ejpam-5974	25	13	networks	network	NOUN
ejpam-5974	25	14	with	with	ADP
ejpam-5974	25	15	respect	respect	NOUN
ejpam-5974	25	16	to	to	ADP
ejpam-5974	25	17	their	their	PRON
ejpam-5974	25	18	density	density	NOUN
ejpam-5974	25	19	can	can	AUX
ejpam-5974	25	20	reveal	reveal	VERB
ejpam-5974	25	21	important	important	ADJ
ejpam-5974	25	22	functional	functional	ADJ
ejpam-5974	25	23	modules	module	NOUN
ejpam-5974	25	24	or	or	CCONJ
ejpam-5974	25	25	pathways	pathway	NOUN
ejpam-5974	25	26	.	.	PUNCT
ejpam-5974	26	1	in	in	ADP
ejpam-5974	26	2	transportation	transportation	NOUN
ejpam-5974	26	3	and	and	CCONJ
ejpam-5974	26	4	communication	communication	NOUN
ejpam-5974	26	5	networks	network	NOUN
ejpam-5974	26	6	,	,	PUNCT
ejpam-5974	26	7	understanding	understand	VERB
ejpam-5974	26	8	how	how	SCONJ
ejpam-5974	26	9	dense	dense	ADJ
ejpam-5974	26	10	certain	certain	ADJ
ejpam-5974	26	11	regions	region	NOUN
ejpam-5974	26	12	are	be	AUX
ejpam-5974	26	13	can	can	AUX
ejpam-5974	26	14	influence	influence	VERB
ejpam-5974	26	15	optimization	optimization	NOUN
ejpam-5974	26	16	,	,	PUNCT
ejpam-5974	26	17	reliability	reliability	NOUN
ejpam-5974	26	18	,	,	PUNCT
ejpam-5974	26	19	and	and	CCONJ
ejpam-5974	26	20	infrastructure	infrastructure	NOUN
ejpam-5974	26	21	planning	planning	NOUN
ejpam-5974	26	22	.	.	PUNCT
ejpam-5974	27	1	in	in	ADP
ejpam-5974	27	2	cybersecurity	cybersecurity	NOUN
ejpam-5974	27	3	,	,	PUNCT
ejpam-5974	27	4	analyzing	analyze	VERB
ejpam-5974	27	5	the	the	DET
ejpam-5974	27	6	density	density	NOUN
ejpam-5974	27	7	of	of	ADP
ejpam-5974	27	8	connections	connection	NOUN
ejpam-5974	27	9	can	can	AUX
ejpam-5974	27	10	help	help	VERB
ejpam-5974	27	11	detect	detect	VERB
ejpam-5974	27	12	anomalous	anomalous	ADJ
ejpam-5974	27	13	patterns	pattern	NOUN
ejpam-5974	27	14	or	or	CCONJ
ejpam-5974	27	15	vulnerabilities	vulnerability	NOUN
ejpam-5974	27	16	.	.	PUNCT
ejpam-5974	28	1	by	by	ADP
ejpam-5974	28	2	evaluating	evaluate	VERB
ejpam-5974	28	3	a	a	DET
ejpam-5974	28	4	network	network	NOUN
ejpam-5974	28	5	’s	’s	PART
ejpam-5974	28	6	density	density	NOUN
ejpam-5974	28	7	,	,	PUNCT
ejpam-5974	28	8	researchers	researcher	NOUN
ejpam-5974	28	9	are	be	AUX
ejpam-5974	28	10	better	well	ADV
ejpam-5974	28	11	equipped	equip	VERB
ejpam-5974	28	12	to	to	PART
ejpam-5974	28	13	understand	understand	VERB
ejpam-5974	28	14	and	and	CCONJ
ejpam-5974	28	15	visualize	visualize	VERB
ejpam-5974	28	16	its	its	PRON
ejpam-5974	28	17	underlying	underlie	VERB
ejpam-5974	28	18	structure	structure	NOUN
ejpam-5974	28	19	,	,	PUNCT
ejpam-5974	28	20	helping	help	VERB
ejpam-5974	28	21	to	to	PART
ejpam-5974	28	22	identify	identify	VERB
ejpam-5974	28	23	trends	trend	NOUN
ejpam-5974	28	24	,	,	PUNCT
ejpam-5974	28	25	cluster	cluster	NOUN
ejpam-5974	28	26	formations	formation	NOUN
ejpam-5974	28	27	,	,	PUNCT
ejpam-5974	28	28	community	community	NOUN
ejpam-5974	28	29	detection	detection	NOUN
ejpam-5974	28	30	,	,	PUNCT
ejpam-5974	28	31	bottlenecks	bottleneck	NOUN
ejpam-5974	28	32	,	,	PUNCT
ejpam-5974	28	33	and	and	CCONJ
ejpam-5974	28	34	other	other	ADJ
ejpam-5974	28	35	critical	critical	ADJ
ejpam-5974	28	36	attributes	attribute	NOUN
ejpam-5974	28	37	.	.	PUNCT
ejpam-5974	29	1	density	density	NOUN
ejpam-5974	29	2	also	also	ADV
ejpam-5974	29	3	influences	influence	VERB
ejpam-5974	29	4	the	the	DET
ejpam-5974	29	5	choice	choice	NOUN
ejpam-5974	29	6	of	of	ADP
ejpam-5974	29	7	algorithms	algorithm	NOUN
ejpam-5974	29	8	for	for	ADP
ejpam-5974	29	9	tasks	task	NOUN
ejpam-5974	29	10	such	such	ADJ
ejpam-5974	29	11	as	as	ADP
ejpam-5974	29	12	traversal	traversal	NOUN
ejpam-5974	29	13	,	,	PUNCT
ejpam-5974	29	14	shortest	short	ADJ
ejpam-5974	29	15	path	path	NOUN
ejpam-5974	29	16	computation	computation	NOUN
ejpam-5974	29	17	,	,	PUNCT
ejpam-5974	29	18	and	and	CCONJ
ejpam-5974	29	19	centrality	centrality	NOUN
ejpam-5974	29	20	measures	measure	NOUN
ejpam-5974	29	21	,	,	PUNCT
ejpam-5974	29	22	and	and	CCONJ
ejpam-5974	29	23	is	be	AUX
ejpam-5974	29	24	closely	closely	ADV
ejpam-5974	29	25	linked	link	VERB
ejpam-5974	29	26	to	to	ADP
ejpam-5974	29	27	concepts	concept	NOUN
ejpam-5974	29	28	such	such	ADJ
ejpam-5974	29	29	as	as	ADP
ejpam-5974	29	30	graph	graph	NOUN
ejpam-5974	29	31	sparsification	sparsification	NOUN
ejpam-5974	29	32	and	and	CCONJ
ejpam-5974	29	33	complexity	complexity	NOUN
ejpam-5974	29	34	analysis	analysis	NOUN
ejpam-5974	29	35	.	.	PUNCT
ejpam-5974	30	1	this	this	DET
ejpam-5974	30	2	article	article	NOUN
ejpam-5974	30	3	focuses	focus	VERB
ejpam-5974	30	4	on	on	ADP
ejpam-5974	30	5	nine	nine	NUM
ejpam-5974	30	6	classical	classical	ADJ
ejpam-5974	30	7	graph	graph	NOUN
ejpam-5974	30	8	that	that	PRON
ejpam-5974	30	9	routinely	routinely	ADV
ejpam-5974	30	10	appear	appear	VERB
ejpam-5974	30	11	in	in	ADP
ejpam-5974	30	12	external	external	ADJ
ejpam-5974	30	13	,	,	PUNCT
ejpam-5974	30	14	labeling	labeling	NOUN
ejpam-5974	30	15	,	,	PUNCT
ejpam-5974	30	16	and	and	CCONJ
ejpam-5974	30	17	network	network	NOUN
ejpam-5974	30	18	analysis	analysis	NOUN
ejpam-5974	30	19	problems	problem	NOUN
ejpam-5974	30	20	:	:	PUNCT
ejpam-5974	30	21	barbell	barbell	NOUN
ejpam-5974	30	22	,	,	PUNCT
ejpam-5974	30	23	friendship	friendship	NOUN
ejpam-5974	30	24	,	,	PUNCT
ejpam-5974	30	25	sunlet	sunlet	NOUN
ejpam-5974	30	26	,	,	PUNCT
ejpam-5974	30	27	banana	banana	NOUN
ejpam-5974	30	28	,	,	PUNCT
ejpam-5974	30	29	lollipop	lollipop	NOUN
ejpam-5974	30	30	,	,	PUNCT
ejpam-5974	30	31	gear	gear	NOUN
ejpam-5974	30	32	,	,	PUNCT
ejpam-5974	30	33	tadpole	tadpole	NOUN
ejpam-5974	30	34	,	,	PUNCT
ejpam-5974	30	35	wheel	wheel	NOUN
ejpam-5974	30	36	,	,	PUNCT
ejpam-5974	30	37	and	and	CCONJ
ejpam-5974	30	38	fan	fan	NOUN
ejpam-5974	30	39	graph	graph	NOUN
ejpam-5974	30	40	.	.	PUNCT
ejpam-5974	31	1	for	for	ADP
ejpam-5974	31	2	each	each	DET
ejpam-5974	31	3	family	family	NOUN
ejpam-5974	31	4	we	we	PRON
ejpam-5974	31	5	determined	determine	VERB
ejpam-5974	31	6	closed	closed	ADJ
ejpam-5974	31	7	form	form	NOUN
ejpam-5974	31	8	expression	expression	NOUN
ejpam-5974	31	9	for	for	ADP
ejpam-5974	31	10	|v	|v	PROPN
ejpam-5974	31	11	(	(	PUNCT
ejpam-5974	31	12	g)|	g)|	NOUN
ejpam-5974	31	13	and	and	CCONJ
ejpam-5974	31	14	|e(g)|	|e(g)|	NOUN
ejpam-5974	31	15	,	,	PUNCT
ejpam-5974	31	16	substituting	substitute	VERB
ejpam-5974	31	17	the	the	PRON
ejpam-5974	31	18	into	into	ADP
ejpam-5974	31	19	d(g	d(g	PROPN
ejpam-5974	31	20	)	)	PUNCT
ejpam-5974	31	21	,	,	PUNCT
ejpam-5974	31	22	and	and	CCONJ
ejpam-5974	31	23	simplify	simplify	VERB
ejpam-5974	31	24	to	to	PART
ejpam-5974	31	25	obtain	obtain	VERB
ejpam-5974	31	26	an	an	DET
ejpam-5974	31	27	exact	exact	ADJ
ejpam-5974	31	28	formula	formula	NOUN
ejpam-5974	31	29	.	.	PUNCT
ejpam-5974	32	1	also	also	ADV
ejpam-5974	32	2	,	,	PUNCT
ejpam-5974	32	3	their	their	PRON
ejpam-5974	32	4	respective	respective	ADJ
ejpam-5974	32	5	mycielski	mycielski	ADJ
ejpam-5974	32	6	graphs	graph	NOUN
ejpam-5974	32	7	which	which	PRON
ejpam-5974	32	8	are	be	AUX
ejpam-5974	32	9	of	of	ADP
ejpam-5974	32	10	particular	particular	ADJ
ejpam-5974	32	11	interest	interest	NOUN
ejpam-5974	32	12	due	due	ADP
ejpam-5974	32	13	to	to	ADP
ejpam-5974	32	14	their	their	PRON
ejpam-5974	32	15	ability	ability	NOUN
ejpam-5974	32	16	to	to	PART
ejpam-5974	32	17	increase	increase	VERB
ejpam-5974	32	18	chromatic	chromatic	ADJ
ejpam-5974	32	19	number	number	NOUN
ejpam-5974	32	20	without	without	ADP
ejpam-5974	32	21	introducing	introduce	VERB
ejpam-5974	32	22	new	new	ADJ
ejpam-5974	32	23	triangles	triangle	NOUN
ejpam-5974	32	24	are	be	AUX
ejpam-5974	32	25	also	also	ADV
ejpam-5974	32	26	examined	examine	VERB
ejpam-5974	32	27	to	to	PART
ejpam-5974	32	28	explore	explore	VERB
ejpam-5974	32	29	how	how	SCONJ
ejpam-5974	32	30	density	density	NOUN
ejpam-5974	32	31	evolves	evolve	VERB
ejpam-5974	32	32	under	under	ADP
ejpam-5974	32	33	graph	graph	NOUN
ejpam-5974	32	34	transformations	transformation	NOUN
ejpam-5974	32	35	.	.	PUNCT
ejpam-5974	33	1	future	future	ADJ
ejpam-5974	33	2	sections	section	NOUN
ejpam-5974	33	3	of	of	ADP
ejpam-5974	33	4	this	this	DET
ejpam-5974	33	5	paper	paper	NOUN
ejpam-5974	33	6	will	will	AUX
ejpam-5974	33	7	explore	explore	VERB
ejpam-5974	33	8	formal	formal	ADJ
ejpam-5974	33	9	definitions	definition	NOUN
ejpam-5974	33	10	of	of	ADP
ejpam-5974	33	11	graph	graph	NOUN
ejpam-5974	33	12	density	density	NOUN
ejpam-5974	33	13	,	,	PUNCT
ejpam-5974	33	14	methods	method	NOUN
ejpam-5974	33	15	of	of	ADP
ejpam-5974	33	16	computation	computation	NOUN
ejpam-5974	33	17	,	,	PUNCT
ejpam-5974	33	18	and	and	CCONJ
ejpam-5974	33	19	comparative	comparative	ADJ
ejpam-5974	33	20	results	result	NOUN
ejpam-5974	33	21	based	base	VERB
ejpam-5974	33	22	on	on	ADP
ejpam-5974	33	23	the	the	DET
ejpam-5974	33	24	selected	select	VERB
ejpam-5974	33	25	graph	graph	NOUN
ejpam-5974	33	26	families	family	NOUN
ejpam-5974	33	27	.	.	PUNCT
ejpam-5974	34	1	2	2	X
ejpam-5974	34	2	.	.	X
ejpam-5974	34	3	terminology	terminology	NOUN
ejpam-5974	34	4	and	and	CCONJ
ejpam-5974	34	5	notation	notation	NOUN
ejpam-5974	34	6	a	a	DET
ejpam-5974	34	7	graph	graph	NOUN
ejpam-5974	34	8	g	g	PROPN
ejpam-5974	34	9	is	be	AUX
ejpam-5974	34	10	a	a	DET
ejpam-5974	34	11	finite	finite	NOUN
ejpam-5974	34	12	nonempty	nonempty	ADV
ejpam-5974	34	13	set	set	VERB
ejpam-5974	34	14	v	v	NOUN
ejpam-5974	34	15	of	of	ADP
ejpam-5974	34	16	objects	object	NOUN
ejpam-5974	34	17	called	call	VERB
ejpam-5974	34	18	vertices	vertex	NOUN
ejpam-5974	34	19	together	together	ADV
ejpam-5974	34	20	with	with	ADP
ejpam-5974	34	21	a	a	DET
ejpam-5974	34	22	possibly	possibly	ADV
ejpam-5974	34	23	empty	empty	ADJ
ejpam-5974	34	24	set	set	ADJ
ejpam-5974	34	25	e	e	NOUN
ejpam-5974	34	26	of	of	ADP
ejpam-5974	34	27	2	2	NUM
ejpam-5974	34	28	-	-	PUNCT
ejpam-5974	34	29	element	element	NOUN
ejpam-5974	34	30	sets	set	NOUN
ejpam-5974	34	31	of	of	ADP
ejpam-5974	34	32	v	v	NOUN
ejpam-5974	34	33	called	call	VERB
ejpam-5974	34	34	edges	edge	NOUN
ejpam-5974	34	35	.	.	PUNCT
ejpam-5974	35	1	to	to	PART
ejpam-5974	35	2	indicate	indicate	VERB
ejpam-5974	35	3	that	that	SCONJ
ejpam-5974	35	4	a	a	DET
ejpam-5974	35	5	graph	graph	NOUN
ejpam-5974	35	6	g	g	NOUN
ejpam-5974	35	7	has	have	VERB
ejpam-5974	35	8	vertex	vertex	NOUN
ejpam-5974	35	9	set	set	VERB
ejpam-5974	35	10	v	v	NOUN
ejpam-5974	35	11	and	and	CCONJ
ejpam-5974	35	12	edge	edge	NOUN
ejpam-5974	35	13	set	set	ADJ
ejpam-5974	35	14	e	e	NOUN
ejpam-5974	35	15	,	,	PUNCT
ejpam-5974	35	16	we	we	PRON
ejpam-5974	35	17	write	write	VERB
ejpam-5974	35	18	g	g	PROPN
ejpam-5974	35	19	=	=	SYM
ejpam-5974	35	20	(	(	PUNCT
ejpam-5974	35	21	v	v	NOUN
ejpam-5974	35	22	,	,	PUNCT
ejpam-5974	35	23	e	e	NOUN
ejpam-5974	35	24	)	)	PUNCT
ejpam-5974	35	25	.	.	PUNCT
ejpam-5974	36	1	to	to	PART
ejpam-5974	36	2	emphasize	emphasize	VERB
ejpam-5974	36	3	that	that	DET
ejpam-5974	36	4	v	v	NOUN
ejpam-5974	36	5	and	and	CCONJ
ejpam-5974	36	6	e	e	NOUN
ejpam-5974	36	7	are	be	AUX
ejpam-5974	36	8	the	the	DET
ejpam-5974	36	9	vertex	vertex	NOUN
ejpam-5974	36	10	set	set	NOUN
ejpam-5974	36	11	and	and	CCONJ
ejpam-5974	36	12	edge	edge	NOUN
ejpam-5974	36	13	set	set	NOUN
ejpam-5974	36	14	of	of	ADP
ejpam-5974	36	15	a	a	DET
ejpam-5974	36	16	graph	graph	NOUN
ejpam-5974	36	17	g	g	NOUN
ejpam-5974	36	18	,	,	PUNCT
ejpam-5974	36	19	we	we	PRON
ejpam-5974	36	20	often	often	ADV
ejpam-5974	36	21	write	write	VERB
ejpam-5974	36	22	v	v	NOUN
ejpam-5974	36	23	as	as	ADP
ejpam-5974	36	24	v	v	NOUN
ejpam-5974	36	25	(	(	PUNCT
ejpam-5974	36	26	g	g	NOUN
ejpam-5974	36	27	)	)	PUNCT
ejpam-5974	36	28	and	and	CCONJ
ejpam-5974	36	29	e	e	NOUN
ejpam-5974	36	30	as	as	ADP
ejpam-5974	36	31	e(g	e(g	PROPN
ejpam-5974	36	32	)	)	PUNCT
ejpam-5974	36	33	.	.	PUNCT
ejpam-5974	37	1	each	each	DET
ejpam-5974	37	2	edge	edge	NOUN
ejpam-5974	37	3	{	{	PUNCT
ejpam-5974	37	4	u	u	NOUN
ejpam-5974	37	5	,	,	PUNCT
ejpam-5974	37	6	v	v	NOUN
ejpam-5974	37	7	}	}	PUNCT
ejpam-5974	37	8	of	of	ADP
ejpam-5974	37	9	g	g	PROPN
ejpam-5974	37	10	is	be	AUX
ejpam-5974	37	11	usually	usually	ADV
ejpam-5974	37	12	denoted	denote	VERB
ejpam-5974	37	13	by	by	ADP
ejpam-5974	37	14	uv	uv	NOUN
ejpam-5974	37	15	or	or	CCONJ
ejpam-5974	37	16	vu	vu	NOUN
ejpam-5974	37	17	.	.	PUNCT
ejpam-5974	38	1	the	the	DET
ejpam-5974	38	2	number	number	NOUN
ejpam-5974	38	3	of	of	ADP
ejpam-5974	38	4	vertices	vertex	NOUN
ejpam-5974	38	5	in	in	ADP
ejpam-5974	38	6	a	a	DET
ejpam-5974	38	7	graph	graph	NOUN
ejpam-5974	38	8	g	g	NOUN
ejpam-5974	38	9	is	be	AUX
ejpam-5974	38	10	the	the	DET
ejpam-5974	38	11	order	order	NOUN
ejpam-5974	38	12	of	of	ADP
ejpam-5974	38	13	g	g	NOUN
ejpam-5974	38	14	and	and	CCONJ
ejpam-5974	38	15	the	the	DET
ejpam-5974	38	16	number	number	NOUN
ejpam-5974	38	17	of	of	ADP
ejpam-5974	38	18	edges	edge	NOUN
ejpam-5974	38	19	is	be	AUX
ejpam-5974	38	20	the	the	DET
ejpam-5974	38	21	size	size	NOUN
ejpam-5974	38	22	of	of	ADP
ejpam-5974	38	23	g.	g.	PROPN
ejpam-5974	38	24	the	the	DET
ejpam-5974	38	25	degree	degree	NOUN
ejpam-5974	38	26	of	of	ADP
ejpam-5974	38	27	a	a	DET
ejpam-5974	38	28	vertex	vertex	NOUN
ejpam-5974	38	29	v	v	NOUN
ejpam-5974	38	30	is	be	AUX
ejpam-5974	38	31	denoted	denote	VERB
ejpam-5974	38	32	by	by	ADP
ejpam-5974	38	33	deg(v	deg(v	PROPN
ejpam-5974	38	34	)	)	PUNCT
ejpam-5974	38	35	and	and	CCONJ
ejpam-5974	38	36	the	the	DET
ejpam-5974	38	37	minimum	minimum	NOUN
ejpam-5974	38	38	degree	degree	NOUN
ejpam-5974	38	39	of	of	ADP
ejpam-5974	38	40	g	g	PROPN
ejpam-5974	38	41	is	be	AUX
ejpam-5974	38	42	denoted	denote	VERB
ejpam-5974	38	43	by	by	ADP
ejpam-5974	38	44	δ(g	δ(g	NOUN
ejpam-5974	38	45	)	)	PUNCT
ejpam-5974	38	46	and	and	CCONJ
ejpam-5974	38	47	the	the	DET
ejpam-5974	38	48	maximum	maximum	ADJ
ejpam-5974	38	49	degree	degree	NOUN
ejpam-5974	38	50	of	of	ADP
ejpam-5974	38	51	g	g	PROPN
ejpam-5974	38	52	is	be	AUX
ejpam-5974	38	53	denoted	denote	VERB
ejpam-5974	38	54	by	by	ADP
ejpam-5974	38	55	∆(g	∆(g	PROPN
ejpam-5974	38	56	)	)	PUNCT
ejpam-5974	39	1	[	[	X
ejpam-5974	39	2	6	6	NUM
ejpam-5974	39	3	]	]	PUNCT
ejpam-5974	39	4	.	.	PUNCT
ejpam-5974	40	1	an	an	DET
ejpam-5974	40	2	empty	empty	ADJ
ejpam-5974	40	3	graph	graph	NOUN
ejpam-5974	40	4	of	of	ADP
ejpam-5974	40	5	order	order	NOUN
ejpam-5974	40	6	n	n	NOUN
ejpam-5974	40	7	is	be	AUX
ejpam-5974	40	8	graph	graph	VERB
ejpam-5974	40	9	with	with	ADP
ejpam-5974	40	10	n	n	ADP
ejpam-5974	40	11	vertices	vertex	NOUN
ejpam-5974	40	12	where	where	SCONJ
ejpam-5974	40	13	in	in	ADP
ejpam-5974	40	14	every	every	DET
ejpam-5974	40	15	pair	pair	NOUN
ejpam-5974	40	16	of	of	ADP
ejpam-5974	40	17	distinct	distinct	ADJ
ejpam-5974	40	18	vertices	vertex	NOUN
ejpam-5974	40	19	are	be	AUX
ejpam-5974	40	20	not	not	PART
ejpam-5974	40	21	adjacent	adjacent	ADJ
ejpam-5974	40	22	[	[	X
ejpam-5974	40	23	6	6	NUM
ejpam-5974	40	24	]	]	PUNCT
ejpam-5974	40	25	.	.	PUNCT
ejpam-5974	41	1	for	for	ADP
ejpam-5974	41	2	an	an	DET
ejpam-5974	41	3	integer	integer	NOUN
ejpam-5974	41	4	n	n	PRON
ejpam-5974	41	5	≥	≥	NOUN
ejpam-5974	41	6	1	1	NUM
ejpam-5974	41	7	,	,	PUNCT
ejpam-5974	41	8	the	the	DET
ejpam-5974	41	9	path	path	NOUN
ejpam-5974	41	10	graph	graph	NOUN
ejpam-5974	41	11	pn	pn	PROPN
ejpam-5974	41	12	is	be	AUX
ejpam-5974	41	13	a	a	DET
ejpam-5974	41	14	graph	graph	NOUN
ejpam-5974	41	15	of	of	ADP
ejpam-5974	41	16	order	order	NOUN
ejpam-5974	41	17	n	n	NOUN
ejpam-5974	41	18	and	and	CCONJ
ejpam-5974	41	19	size	size	NOUN
ejpam-5974	41	20	n−1	n−1	PROPN
ejpam-5974	41	21	whose	whose	DET
ejpam-5974	41	22	vertices	vertex	NOUN
ejpam-5974	41	23	can	can	AUX
ejpam-5974	41	24	be	be	AUX
ejpam-5974	41	25	labeled	label	VERB
ejpam-5974	41	26	as	as	ADP
ejpam-5974	41	27	v0	v0	NOUN
ejpam-5974	41	28	,	,	PUNCT
ejpam-5974	41	29	v1	v1	NOUN
ejpam-5974	41	30	,	,	PUNCT
ejpam-5974	41	31	v2	v2	PROPN
ejpam-5974	41	32	,	,	PUNCT
ejpam-5974	41	33	...	...	PUNCT
ejpam-5974	41	34	,	,	PUNCT
ejpam-5974	41	35	vn−1	vn−1	ADJ
ejpam-5974	41	36	and	and	CCONJ
ejpam-5974	41	37	whose	whose	DET
ejpam-5974	41	38	edges	edge	NOUN
ejpam-5974	41	39	are	be	AUX
ejpam-5974	41	40	vi	vi	ADJ
ejpam-5974	41	41	,	,	PUNCT
ejpam-5974	41	42	vi+1	vi+1	NOUN
ejpam-5974	41	43	for	for	ADP
ejpam-5974	41	44	i	i	PROPN
ejpam-5974	41	45	=	=	NOUN
ejpam-5974	41	46	1	1	NUM
ejpam-5974	41	47	,	,	PUNCT
ejpam-5974	41	48	2	2	NUM
ejpam-5974	41	49	,	,	PUNCT
ejpam-5974	41	50	3	3	NUM
ejpam-5974	41	51	,	,	PUNCT
ejpam-5974	41	52	...	...	PUNCT
ejpam-5974	41	53	,	,	PUNCT
ejpam-5974	41	54	n−2	n−2	PROPN
ejpam-5974	41	55	r.	r.	PROPN
ejpam-5974	41	56	sango	sango	PROPN
ejpam-5974	41	57	,	,	PUNCT
ejpam-5974	41	58	i.	i.	NOUN
ejpam-5974	41	59	cabahug	cabahug	PROPN
ejpam-5974	41	60	,	,	PUNCT
ejpam-5974	41	61	jr	jr	PROPN
ejpam-5974	41	62	/	/	SYM
ejpam-5974	41	63	eur	eur	PROPN
ejpam-5974	41	64	.	.	PUNCT
ejpam-5974	42	1	j.	j.	PROPN
ejpam-5974	42	2	pure	pure	PROPN
ejpam-5974	42	3	appl	appl	PROPN
ejpam-5974	42	4	.	.	PROPN
ejpam-5974	42	5	math	math	PROPN
ejpam-5974	42	6	,	,	PUNCT
ejpam-5974	42	7	18	18	NUM
ejpam-5974	42	8	(	(	PUNCT
ejpam-5974	42	9	3	3	NUM
ejpam-5974	42	10	)	)	PUNCT
ejpam-5974	42	11	(	(	PUNCT
ejpam-5974	42	12	2025	2025	NUM
ejpam-5974	42	13	)	)	PUNCT
ejpam-5974	42	14	,	,	PUNCT
ejpam-5974	42	15	5974	5974	NUM
ejpam-5974	42	16	3	3	NUM
ejpam-5974	42	17	of	of	ADP
ejpam-5974	42	18	16	16	NUM
ejpam-5974	43	1	[	[	X
ejpam-5974	43	2	6	6	NUM
ejpam-5974	43	3	]	]	PUNCT
ejpam-5974	43	4	.	.	PUNCT
ejpam-5974	44	1	for	for	ADP
ejpam-5974	44	2	an	an	DET
ejpam-5974	44	3	integer	integer	NOUN
ejpam-5974	44	4	n	n	PRON
ejpam-5974	44	5	≥	≥	NOUN
ejpam-5974	44	6	1	1	NUM
ejpam-5974	44	7	,	,	PUNCT
ejpam-5974	44	8	the	the	DET
ejpam-5974	44	9	path	path	NOUN
ejpam-5974	44	10	graph	graph	NOUN
ejpam-5974	44	11	cn	cn	PROPN
ejpam-5974	44	12	is	be	AUX
ejpam-5974	44	13	a	a	DET
ejpam-5974	44	14	graph	graph	NOUN
ejpam-5974	44	15	of	of	ADP
ejpam-5974	44	16	order	order	NOUN
ejpam-5974	44	17	n	n	NOUN
ejpam-5974	44	18	and	and	CCONJ
ejpam-5974	44	19	size	size	NOUN
ejpam-5974	44	20	n	n	CCONJ
ejpam-5974	44	21	whose	whose	DET
ejpam-5974	44	22	vertices	vertex	NOUN
ejpam-5974	44	23	can	can	AUX
ejpam-5974	44	24	be	be	AUX
ejpam-5974	44	25	labeled	label	VERB
ejpam-5974	44	26	as	as	ADP
ejpam-5974	44	27	v0	v0	NOUN
ejpam-5974	44	28	,	,	PUNCT
ejpam-5974	44	29	v1	v1	NOUN
ejpam-5974	44	30	,	,	PUNCT
ejpam-5974	44	31	v2	v2	PROPN
ejpam-5974	44	32	,	,	PUNCT
ejpam-5974	44	33	...	...	PUNCT
ejpam-5974	44	34	,	,	PUNCT
ejpam-5974	44	35	vn−1	vn−1	ADJ
ejpam-5974	44	36	and	and	CCONJ
ejpam-5974	44	37	whose	whose	DET
ejpam-5974	44	38	edges	edge	NOUN
ejpam-5974	44	39	are	be	AUX
ejpam-5974	44	40	vi	vi	ADJ
ejpam-5974	44	41	,	,	PUNCT
ejpam-5974	44	42	vi+1	vi+1	NOUN
ejpam-5974	44	43	for	for	ADP
ejpam-5974	44	44	i	i	PROPN
ejpam-5974	44	45	=	=	NOUN
ejpam-5974	44	46	1	1	NUM
ejpam-5974	44	47	,	,	PUNCT
ejpam-5974	44	48	2	2	NUM
ejpam-5974	44	49	,	,	PUNCT
ejpam-5974	44	50	3	3	NUM
ejpam-5974	44	51	,	,	PUNCT
ejpam-5974	44	52	...	...	PUNCT
ejpam-5974	44	53	,	,	PUNCT
ejpam-5974	44	54	n−	n−	NOUN
ejpam-5974	44	55	2	2	NUM
ejpam-5974	44	56	[	[	X
ejpam-5974	44	57	6	6	NUM
ejpam-5974	44	58	]	]	PUNCT
ejpam-5974	44	59	.	.	PUNCT
ejpam-5974	45	1	for	for	ADP
ejpam-5974	45	2	n	n	PRON
ejpam-5974	45	3	≥	≥	NUM
ejpam-5974	45	4	3	3	NUM
ejpam-5974	45	5	,	,	PUNCT
ejpam-5974	45	6	the	the	DET
ejpam-5974	45	7	fan	fan	NOUN
ejpam-5974	45	8	graph	graph	NOUN
ejpam-5974	45	9	fn	fn	NOUN
ejpam-5974	45	10	of	of	ADP
ejpam-5974	45	11	order	order	NOUN
ejpam-5974	45	12	n+1	n+1	ADV
ejpam-5974	45	13	is	be	AUX
ejpam-5974	45	14	a	a	DET
ejpam-5974	45	15	graph	graph	NOUN
ejpam-5974	45	16	obtained	obtain	VERB
ejpam-5974	45	17	by	by	ADP
ejpam-5974	45	18	connecting	connect	VERB
ejpam-5974	45	19	a	a	DET
ejpam-5974	45	20	new	new	ADJ
ejpam-5974	45	21	vertex	vertex	NOUN
ejpam-5974	45	22	v	v	NOUN
ejpam-5974	45	23	to	to	ADP
ejpam-5974	45	24	each	each	DET
ejpam-5974	45	25	vertex	vertex	NOUN
ejpam-5974	45	26	of	of	ADP
ejpam-5974	45	27	the	the	DET
ejpam-5974	45	28	path	path	NOUN
ejpam-5974	45	29	pn	pn	PROPN
ejpam-5974	46	1	[	[	X
ejpam-5974	46	2	7	7	NUM
ejpam-5974	46	3	]	]	PUNCT
ejpam-5974	46	4	.	.	PUNCT
ejpam-5974	47	1	tadpole	tadpole	PROPN
ejpam-5974	47	2	graph	graph	NOUN
ejpam-5974	47	3	(	(	PUNCT
ejpam-5974	47	4	tm	tm	NOUN
ejpam-5974	47	5	,	,	PUNCT
ejpam-5974	47	6	n	n	CCONJ
ejpam-5974	47	7	)	)	PUNCT
ejpam-5974	47	8	is	be	AUX
ejpam-5974	47	9	defined	define	VERB
ejpam-5974	47	10	as	as	ADP
ejpam-5974	47	11	a	a	DET
ejpam-5974	47	12	graph	graph	NOUN
ejpam-5974	47	13	obtained	obtain	VERB
ejpam-5974	47	14	by	by	ADP
ejpam-5974	47	15	combining	combine	VERB
ejpam-5974	47	16	a	a	DET
ejpam-5974	47	17	vertex	vertex	NOUN
ejpam-5974	47	18	of	of	ADP
ejpam-5974	47	19	cycle	cycle	NOUN
ejpam-5974	47	20	cm	cm	NOUN
ejpam-5974	47	21	with	with	ADP
ejpam-5974	47	22	one	one	NUM
ejpam-5974	47	23	of	of	ADP
ejpam-5974	47	24	leaf	leaf	NOUN
ejpam-5974	47	25	of	of	ADP
ejpam-5974	47	26	path	path	NOUN
ejpam-5974	47	27	pn	pn	PROPN
ejpam-5974	48	1	[	[	X
ejpam-5974	48	2	8	8	NUM
ejpam-5974	48	3	]	]	PUNCT
ejpam-5974	48	4	.	.	PUNCT
ejpam-5974	49	1	a	a	DET
ejpam-5974	49	2	complete	complete	ADJ
ejpam-5974	49	3	graphof	graphof	ADJ
ejpam-5974	49	4	order	order	NOUN
ejpam-5974	49	5	n	n	PRON
ejpam-5974	49	6	≥	≥	NOUN
ejpam-5974	49	7	2	2	NUM
ejpam-5974	49	8	,	,	PUNCT
ejpam-5974	49	9	denoted	denote	VERB
ejpam-5974	49	10	by	by	ADP
ejpam-5974	49	11	kn	kn	PROPN
ejpam-5974	49	12	,	,	PUNCT
ejpam-5974	49	13	is	be	AUX
ejpam-5974	49	14	a	a	DET
ejpam-5974	49	15	graph	graph	NOUN
ejpam-5974	49	16	with	with	ADP
ejpam-5974	49	17	n	n	ADP
ejpam-5974	49	18	vertices	vertex	NOUN
ejpam-5974	49	19	where	where	SCONJ
ejpam-5974	49	20	in	in	ADP
ejpam-5974	49	21	every	every	DET
ejpam-5974	49	22	pair	pair	NOUN
ejpam-5974	49	23	of	of	ADP
ejpam-5974	49	24	distinct	distinct	ADJ
ejpam-5974	49	25	vertices	vertex	NOUN
ejpam-5974	49	26	are	be	AUX
ejpam-5974	49	27	adjacent	adjacent	ADJ
ejpam-5974	49	28	[	[	X
ejpam-5974	49	29	6	6	NUM
ejpam-5974	49	30	]	]	PUNCT
ejpam-5974	49	31	.	.	PUNCT
ejpam-5974	50	1	barbell	barbell	PROPN
ejpam-5974	50	2	graph	graph	NOUN
ejpam-5974	50	3	bn	bn	PROPN
ejpam-5974	50	4	is	be	AUX
ejpam-5974	50	5	a	a	DET
ejpam-5974	50	6	graph	graph	NOUN
ejpam-5974	50	7	obtained	obtain	VERB
ejpam-5974	50	8	by	by	ADP
ejpam-5974	50	9	connecting	connect	VERB
ejpam-5974	50	10	two	two	NUM
ejpam-5974	50	11	complete	complete	ADJ
ejpam-5974	50	12	graph	graph	NOUN
ejpam-5974	50	13	kn	kn	PROPN
ejpam-5974	50	14	by	by	ADP
ejpam-5974	50	15	an	an	DET
ejpam-5974	50	16	edge	edge	NOUN
ejpam-5974	50	17	[	[	X
ejpam-5974	50	18	8	8	NUM
ejpam-5974	50	19	]	]	PUNCT
ejpam-5974	50	20	.	.	PUNCT
ejpam-5974	51	1	the	the	DET
ejpam-5974	51	2	(	(	PUNCT
ejpam-5974	51	3	m	m	PROPN
ejpam-5974	51	4	,	,	PUNCT
ejpam-5974	51	5	n)−lollipop	n)−lollipop	ADJ
ejpam-5974	51	6	graph	graph	NOUN
ejpam-5974	51	7	denoted	denote	VERB
ejpam-5974	51	8	by	by	ADP
ejpam-5974	51	9	lm	lm	NOUN
ejpam-5974	51	10	,	,	PUNCT
ejpam-5974	51	11	n	n	PRON
ejpam-5974	51	12	is	be	AUX
ejpam-5974	51	13	a	a	DET
ejpam-5974	51	14	graph	graph	NOUN
ejpam-5974	51	15	obtained	obtain	VERB
ejpam-5974	51	16	by	by	ADP
ejpam-5974	51	17	joining	join	VERB
ejpam-5974	51	18	a	a	DET
ejpam-5974	51	19	complete	complete	ADJ
ejpam-5974	51	20	graph	graph	NOUN
ejpam-5974	51	21	km	km	NOUN
ejpam-5974	51	22	to	to	ADP
ejpam-5974	51	23	a	a	DET
ejpam-5974	51	24	path	path	NOUN
ejpam-5974	51	25	graph	graph	NOUN
ejpam-5974	51	26	pn	pn	NOUN
ejpam-5974	51	27	with	with	ADP
ejpam-5974	51	28	a	a	DET
ejpam-5974	51	29	bridge	bridge	NOUN
ejpam-5974	51	30	[	[	X
ejpam-5974	51	31	9	9	NUM
ejpam-5974	51	32	]	]	PUNCT
ejpam-5974	51	33	.	.	PUNCT
ejpam-5974	52	1	the	the	DET
ejpam-5974	52	2	friendship	friendship	NOUN
ejpam-5974	52	3	graph	graph	NOUN
ejpam-5974	52	4	,	,	PUNCT
ejpam-5974	52	5	denoted	denote	VERB
ejpam-5974	52	6	by	by	ADP
ejpam-5974	52	7	frn	frn	PROPN
ejpam-5974	52	8	is	be	AUX
ejpam-5974	52	9	a	a	DET
ejpam-5974	52	10	set	set	NOUN
ejpam-5974	52	11	of	of	ADP
ejpam-5974	52	12	n	n	DET
ejpam-5974	52	13	triangle	triangle	NOUN
ejpam-5974	52	14	having	have	VERB
ejpam-5974	52	15	a	a	DET
ejpam-5974	52	16	common	common	ADJ
ejpam-5974	52	17	vertex	vertex	NOUN
ejpam-5974	52	18	[	[	X
ejpam-5974	52	19	10	10	NUM
ejpam-5974	52	20	]	]	PUNCT
ejpam-5974	52	21	.	.	PUNCT
ejpam-5974	53	1	the	the	DET
ejpam-5974	53	2	sunlet	sunlet	NOUN
ejpam-5974	53	3	graph	graph	NOUN
ejpam-5974	53	4	sn	sn	PROPN
ejpam-5974	53	5	is	be	AUX
ejpam-5974	53	6	the	the	DET
ejpam-5974	53	7	graph	graph	NOUN
ejpam-5974	53	8	on	on	ADP
ejpam-5974	53	9	2n	2n	NUM
ejpam-5974	53	10	vertices	vertex	NOUN
ejpam-5974	53	11	obtaining	obtain	VERB
ejpam-5974	53	12	by	by	ADP
ejpam-5974	53	13	attaching	attach	VERB
ejpam-5974	53	14	n	n	DET
ejpam-5974	53	15	pendant	pendant	ADJ
ejpam-5974	53	16	edges	edge	NOUN
ejpam-5974	53	17	to	to	ADP
ejpam-5974	53	18	a	a	DET
ejpam-5974	53	19	cycle	cycle	NOUN
ejpam-5974	53	20	graph	graph	NOUN
ejpam-5974	53	21	cn	cn	PROPN
ejpam-5974	54	1	[	[	X
ejpam-5974	54	2	11	11	NUM
ejpam-5974	54	3	]	]	PUNCT
ejpam-5974	54	4	.	.	PUNCT
ejpam-5974	55	1	a	a	DET
ejpam-5974	55	2	banana	banana	NOUN
ejpam-5974	55	3	tree	tree	NOUN
ejpam-5974	55	4	graph	graph	NOUN
ejpam-5974	55	5	bm	bm	PROPN
ejpam-5974	55	6	,	,	PUNCT
ejpam-5974	55	7	n	n	PROPN
ejpam-5974	55	8	is	be	AUX
ejpam-5974	55	9	a	a	DET
ejpam-5974	55	10	graph	graph	NOUN
ejpam-5974	55	11	of	of	ADP
ejpam-5974	55	12	order	order	NOUN
ejpam-5974	55	13	mn	mn	PROPN
ejpam-5974	55	14	+	+	ADV
ejpam-5974	55	15	1	1	NUM
ejpam-5974	55	16	obtained	obtain	VERB
ejpam-5974	55	17	by	by	ADP
ejpam-5974	55	18	connecting	connect	VERB
ejpam-5974	55	19	one	one	NUM
ejpam-5974	55	20	leaf	leaf	NOUN
ejpam-5974	55	21	on	on	ADP
ejpam-5974	55	22	each	each	PRON
ejpam-5974	55	23	of	of	ADP
ejpam-5974	55	24	m	m	PROPN
ejpam-5974	55	25	copies	copy	NOUN
ejpam-5974	55	26	of	of	ADP
ejpam-5974	55	27	star	star	NOUN
ejpam-5974	55	28	k1,n−1	k1,n−1	ADJ
ejpam-5974	55	29	with	with	ADP
ejpam-5974	55	30	a	a	DET
ejpam-5974	55	31	single	single	ADJ
ejpam-5974	55	32	root	root	NOUN
ejpam-5974	55	33	vertex	vertex	NOUN
ejpam-5974	55	34	v	v	NOUN
ejpam-5974	55	35	that	that	PRON
ejpam-5974	55	36	is	be	AUX
ejpam-5974	55	37	distinct	distinct	ADJ
ejpam-5974	55	38	from	from	ADP
ejpam-5974	55	39	all	all	DET
ejpam-5974	55	40	stars	star	NOUN
ejpam-5974	55	41	[	[	X
ejpam-5974	55	42	12	12	NUM
ejpam-5974	55	43	]	]	PUNCT
ejpam-5974	55	44	.	.	PUNCT
ejpam-5974	56	1	for	for	ADP
ejpam-5974	56	2	n	n	PRON
ejpam-5974	56	3	≥	≥	NUM
ejpam-5974	56	4	3	3	NUM
ejpam-5974	56	5	,	,	PUNCT
ejpam-5974	56	6	the	the	DET
ejpam-5974	56	7	wheel	wheel	NOUN
ejpam-5974	56	8	graph	graph	NOUN
ejpam-5974	56	9	,	,	PUNCT
ejpam-5974	56	10	wn	wn	NOUN
ejpam-5974	56	11	of	of	ADP
ejpam-5974	56	12	order	order	NOUN
ejpam-5974	56	13	n+	n+	ADP
ejpam-5974	56	14	1	1	NUM
ejpam-5974	56	15	is	be	AUX
ejpam-5974	56	16	a	a	DET
ejpam-5974	56	17	graph	graph	NOUN
ejpam-5974	56	18	produced	produce	VERB
ejpam-5974	56	19	from	from	ADP
ejpam-5974	56	20	the	the	DET
ejpam-5974	56	21	complete	complete	ADJ
ejpam-5974	56	22	product	product	NOUN
ejpam-5974	56	23	of	of	ADP
ejpam-5974	56	24	an	an	DET
ejpam-5974	56	25	isolated	isolated	ADJ
ejpam-5974	56	26	vertex	vertex	NOUN
ejpam-5974	56	27	and	and	CCONJ
ejpam-5974	56	28	a	a	DET
ejpam-5974	56	29	cycle	cycle	NOUN
ejpam-5974	56	30	cn	cn	PROPN
ejpam-5974	57	1	[	[	X
ejpam-5974	57	2	7	7	NUM
ejpam-5974	57	3	]	]	PUNCT
ejpam-5974	57	4	.	.	PUNCT
ejpam-5974	58	1	a	a	DET
ejpam-5974	58	2	gear	gear	NOUN
ejpam-5974	58	3	graph	graph	NOUN
ejpam-5974	58	4	,	,	PUNCT
ejpam-5974	58	5	denoted	denote	VERB
ejpam-5974	58	6	by	by	ADP
ejpam-5974	58	7	gn	gn	PROPN
ejpam-5974	58	8	,	,	PUNCT
ejpam-5974	58	9	is	be	AUX
ejpam-5974	58	10	obtained	obtain	VERB
ejpam-5974	58	11	from	from	ADP
ejpam-5974	58	12	the	the	DET
ejpam-5974	58	13	wheel	wheel	NOUN
ejpam-5974	58	14	graph	graph	NOUN
ejpam-5974	58	15	by	by	ADP
ejpam-5974	58	16	adding	add	VERB
ejpam-5974	58	17	a	a	DET
ejpam-5974	58	18	vertex	vertex	NOUN
ejpam-5974	58	19	between	between	ADP
ejpam-5974	58	20	every	every	DET
ejpam-5974	58	21	pair	pair	NOUN
ejpam-5974	58	22	of	of	ADP
ejpam-5974	58	23	adjacent	adjacent	ADJ
ejpam-5974	58	24	vertices	vertex	NOUN
ejpam-5974	58	25	of	of	ADP
ejpam-5974	58	26	the	the	DET
ejpam-5974	58	27	cycle	cycle	NOUN
ejpam-5974	58	28	[	[	X
ejpam-5974	58	29	10	10	NUM
ejpam-5974	58	30	]	]	PUNCT
ejpam-5974	58	31	.	.	PUNCT
ejpam-5974	59	1	consider	consider	VERB
ejpam-5974	59	2	a	a	DET
ejpam-5974	59	3	graph	graph	NOUN
ejpam-5974	59	4	g	g	NOUN
ejpam-5974	59	5	with	with	ADP
ejpam-5974	59	6	v	v	NOUN
ejpam-5974	59	7	(	(	PUNCT
ejpam-5974	59	8	g	g	NOUN
ejpam-5974	59	9	)	)	PUNCT
ejpam-5974	59	10	=	=	SYM
ejpam-5974	59	11	v1	v1	NOUN
ejpam-5974	59	12	,	,	PUNCT
ejpam-5974	59	13	v2	v2	PROPN
ejpam-5974	59	14	,	,	PUNCT
ejpam-5974	59	15	v3	v3	PROPN
ejpam-5974	59	16	,	,	PUNCT
ejpam-5974	59	17	...	...	PUNCT
ejpam-5974	59	18	,	,	PUNCT
ejpam-5974	59	19	vn	vn	AUX
ejpam-5974	59	20	.	.	PROPN
ejpam-5974	59	21	apply	apply	VERB
ejpam-5974	59	22	the	the	DET
ejpam-5974	59	23	following	follow	VERB
ejpam-5974	59	24	steps	step	NOUN
ejpam-5974	59	25	to	to	ADP
ejpam-5974	59	26	the	the	DET
ejpam-5974	59	27	graph	graph	NOUN
ejpam-5974	59	28	g	g	NOUN
ejpam-5974	59	29	:	:	PUNCT
ejpam-5974	59	30	(	(	PUNCT
ejpam-5974	59	31	i	i	NOUN
ejpam-5974	59	32	)	)	PUNCT
ejpam-5974	59	33	take	take	VERB
ejpam-5974	59	34	the	the	DET
ejpam-5974	59	35	set	set	NOUN
ejpam-5974	59	36	of	of	ADP
ejpam-5974	59	37	new	new	ADJ
ejpam-5974	59	38	vertices	vertex	NOUN
ejpam-5974	59	39	u	u	NOUN
ejpam-5974	59	40	=	=	NOUN
ejpam-5974	59	41	u1	u1	PROPN
ejpam-5974	59	42	,	,	PUNCT
ejpam-5974	59	43	u2	u2	NOUN
ejpam-5974	59	44	,	,	PUNCT
ejpam-5974	59	45	u3	u3	NOUN
ejpam-5974	59	46	,	,	PUNCT
ejpam-5974	59	47	...	...	PUNCT
ejpam-5974	59	48	,	,	PUNCT
ejpam-5974	59	49	un	un	PROPN
ejpam-5974	59	50	and	and	CCONJ
ejpam-5974	59	51	add	add	VERB
ejpam-5974	59	52	edges	edge	NOUN
ejpam-5974	59	53	from	from	ADP
ejpam-5974	59	54	each	each	DET
ejpam-5974	59	55	vertex	vertex	NOUN
ejpam-5974	59	56	ui	ui	NOUN
ejpam-5974	59	57	of	of	ADP
ejpam-5974	59	58	u	u	PRON
ejpam-5974	59	59	to	to	ADP
ejpam-5974	59	60	the	the	DET
ejpam-5974	59	61	vertices	vertex	NOUN
ejpam-5974	59	62	vj	vj	INTJ
ejpam-5974	59	63	if	if	SCONJ
ejpam-5974	59	64	the	the	DET
ejpam-5974	59	65	corresponding	corresponding	ADJ
ejpam-5974	59	66	vertex	vertex	NOUN
ejpam-5974	59	67	vi	vi	PROPN
ejpam-5974	59	68	is	be	AUX
ejpam-5974	59	69	adjacent	adjacent	ADJ
ejpam-5974	59	70	to	to	ADP
ejpam-5974	59	71	vj	vj	PROPN
ejpam-5974	59	72	in	in	ADP
ejpam-5974	59	73	g.	g.	PROPN
ejpam-5974	59	74	(	(	PUNCT
ejpam-5974	59	75	ii	ii	PROPN
ejpam-5974	59	76	)	)	PUNCT
ejpam-5974	59	77	take	take	VERB
ejpam-5974	59	78	another	another	DET
ejpam-5974	59	79	new	new	ADJ
ejpam-5974	59	80	vertex	vertex	NOUN
ejpam-5974	59	81	w0	w0	NOUN
ejpam-5974	59	82	and	and	CCONJ
ejpam-5974	59	83	add	add	VERB
ejpam-5974	59	84	edges	edge	NOUN
ejpam-5974	59	85	joining	join	VERB
ejpam-5974	59	86	each	each	DET
ejpam-5974	59	87	element	element	NOUN
ejpam-5974	59	88	in	in	ADP
ejpam-5974	59	89	u	u	PROPN
ejpam-5974	59	90	.	.	PUNCT
ejpam-5974	60	1	here	here	ADV
ejpam-5974	60	2	,	,	PUNCT
ejpam-5974	60	3	the	the	DET
ejpam-5974	60	4	new	new	ADJ
ejpam-5974	60	5	graph	graph	NOUN
ejpam-5974	60	6	obtained	obtain	VERB
ejpam-5974	60	7	is	be	AUX
ejpam-5974	60	8	the	the	DET
ejpam-5974	60	9	mycielski	mycielski	ADJ
ejpam-5974	60	10	graph	graph	NOUN
ejpam-5974	60	11	,	,	PUNCT
ejpam-5974	60	12	denoted	denote	VERB
ejpam-5974	60	13	by	by	ADP
ejpam-5974	60	14	µ(g	µ(g	PROPN
ejpam-5974	60	15	)	)	PUNCT
ejpam-5974	60	16	of	of	ADP
ejpam-5974	60	17	graph	graph	NOUN
ejpam-5974	60	18	g	g	PROPN
ejpam-5974	60	19	[	[	X
ejpam-5974	60	20	13	13	NUM
ejpam-5974	60	21	]	]	PUNCT
ejpam-5974	60	22	.	.	PUNCT
ejpam-5974	61	1	let	let	VERB
ejpam-5974	61	2	g	g	PRON
ejpam-5974	61	3	be	be	AUX
ejpam-5974	61	4	undirected	undirected	ADJ
ejpam-5974	61	5	graph	graph	NOUN
ejpam-5974	61	6	with	with	ADP
ejpam-5974	61	7	|v	|v	PROPN
ejpam-5974	61	8	(	(	PUNCT
ejpam-5974	61	9	g)|	g)|	NOUN
ejpam-5974	61	10	and	and	CCONJ
ejpam-5974	61	11	|e(g)|	|e(g)|	PROPN
ejpam-5974	61	12	.	.	PUNCT
ejpam-5974	62	1	the	the	DET
ejpam-5974	62	2	density	density	NOUN
ejpam-5974	62	3	of	of	ADP
ejpam-5974	62	4	the	the	DET
ejpam-5974	62	5	graph	graph	NOUN
ejpam-5974	62	6	g	g	NOUN
ejpam-5974	62	7	,	,	PUNCT
ejpam-5974	62	8	denoted	denote	VERB
ejpam-5974	62	9	as	as	ADP
ejpam-5974	62	10	d(g	d(g	PROPN
ejpam-5974	62	11	)	)	PUNCT
ejpam-5974	62	12	,	,	PUNCT
ejpam-5974	62	13	is	be	AUX
ejpam-5974	62	14	defined	define	VERB
ejpam-5974	62	15	as	as	ADP
ejpam-5974	62	16	the	the	DET
ejpam-5974	62	17	ratio	ratio	NOUN
ejpam-5974	62	18	of	of	ADP
ejpam-5974	62	19	the	the	DET
ejpam-5974	62	20	number	number	NOUN
ejpam-5974	62	21	of	of	ADP
ejpam-5974	62	22	edges	edge	NOUN
ejpam-5974	62	23	in	in	ADP
ejpam-5974	62	24	the	the	DET
ejpam-5974	62	25	graph	graph	NOUN
ejpam-5974	62	26	to	to	ADP
ejpam-5974	62	27	the	the	DET
ejpam-5974	62	28	maximum	maximum	ADJ
ejpam-5974	62	29	possible	possible	ADJ
ejpam-5974	62	30	number	number	NOUN
ejpam-5974	62	31	of	of	ADP
ejpam-5974	62	32	edges	edge	NOUN
ejpam-5974	62	33	between	between	ADP
ejpam-5974	62	34	the	the	DET
ejpam-5974	62	35	number	number	NOUN
ejpam-5974	62	36	of	of	ADP
ejpam-5974	62	37	vertices	vertex	NOUN
ejpam-5974	62	38	[	[	X
ejpam-5974	62	39	14	14	NUM
ejpam-5974	62	40	]	]	PUNCT
ejpam-5974	62	41	.	.	PUNCT
ejpam-5974	63	1	for	for	ADP
ejpam-5974	63	2	the	the	DET
ejpam-5974	63	3	undirected	undirected	ADJ
ejpam-5974	63	4	graph	graph	NOUN
ejpam-5974	63	5	,	,	PUNCT
ejpam-5974	63	6	the	the	DET
ejpam-5974	63	7	density	density	NOUN
ejpam-5974	63	8	is	be	AUX
ejpam-5974	63	9	given	give	VERB
ejpam-5974	63	10	by	by	ADP
ejpam-5974	63	11	the	the	DET
ejpam-5974	63	12	formula	formula	NOUN
ejpam-5974	63	13	:	:	PUNCT
ejpam-5974	63	14	d(g	d(g	X
ejpam-5974	63	15	)	)	PUNCT
ejpam-5974	63	16	=	=	SYM
ejpam-5974	64	1	2|e(g)|	2|e(g)|	NUM
ejpam-5974	64	2	|v	|v	X
ejpam-5974	64	3	(	(	PUNCT
ejpam-5974	64	4	g)|(|v	g)|(|v	X
ejpam-5974	64	5	(	(	PUNCT
ejpam-5974	64	6	g)|	g)|	VERB
ejpam-5974	64	7	−	−	NOUN
ejpam-5974	64	8	1	1	NUM
ejpam-5974	64	9	)	)	PUNCT
ejpam-5974	64	10	.	.	PUNCT
ejpam-5974	65	1	consider	consider	VERB
ejpam-5974	65	2	figure	figure	NOUN
ejpam-5974	65	3	1	1	NUM
ejpam-5974	65	4	,	,	PUNCT
ejpam-5974	65	5	it	it	PRON
ejpam-5974	65	6	shows	show	VERB
ejpam-5974	65	7	that	that	SCONJ
ejpam-5974	65	8	the	the	DET
ejpam-5974	65	9	number	number	NOUN
ejpam-5974	65	10	of	of	ADP
ejpam-5974	65	11	|e(g)|	|e(g)|	PROPN
ejpam-5974	65	12	=	=	PROPN
ejpam-5974	65	13	12	12	NUM
ejpam-5974	65	14	and	and	CCONJ
ejpam-5974	65	15	|v	|v	PROPN
ejpam-5974	65	16	(	(	PUNCT
ejpam-5974	65	17	g)|	g)|	NOUN
ejpam-5974	65	18	=	=	NOUN
ejpam-5974	65	19	11	11	NUM
ejpam-5974	65	20	.	.	PUNCT
ejpam-5974	66	1	clearly	clearly	ADV
ejpam-5974	66	2	,	,	PUNCT
ejpam-5974	66	3	d(g	d(g	PROPN
ejpam-5974	66	4	)	)	PUNCT
ejpam-5974	66	5	=	=	SYM
ejpam-5974	66	6	2|e(g)|	2|e(g)|	NUM
ejpam-5974	66	7	|v	|v	X
ejpam-5974	66	8	(	(	PUNCT
ejpam-5974	66	9	g)|(|v	g)|(|v	X
ejpam-5974	66	10	(	(	PUNCT
ejpam-5974	66	11	g)|	g)|	NOUN
ejpam-5974	66	12	−	−	NOUN
ejpam-5974	66	13	1	1	NUM
ejpam-5974	66	14	)	)	PUNCT
ejpam-5974	66	15	=	=	SYM
ejpam-5974	67	1	2|12|	2|12|	NUM
ejpam-5974	67	2	11(11−	11(11−	NUM
ejpam-5974	67	3	1	1	NUM
ejpam-5974	67	4	)	)	PUNCT
ejpam-5974	67	5	=	=	SYM
ejpam-5974	67	6	24	24	NUM
ejpam-5974	67	7	11(10	11(10	NUM
ejpam-5974	67	8	)	)	PUNCT
ejpam-5974	67	9	=	=	PUNCT
ejpam-5974	68	1	26	26	NUM
ejpam-5974	69	1	110	110	NUM
ejpam-5974	69	2	=	=	SYM
ejpam-5974	69	3	0.218	0.218	NUM
ejpam-5974	69	4	3	3	NUM
ejpam-5974	69	5	.	.	PUNCT
ejpam-5974	69	6	results	result	NOUN
ejpam-5974	69	7	theorem	theorem	VERB
ejpam-5974	69	8	1	1	X
ejpam-5974	69	9	.	.	PUNCT
ejpam-5974	70	1	let	let	VERB
ejpam-5974	70	2	g	g	PRON
ejpam-5974	70	3	be	be	AUX
ejpam-5974	70	4	a	a	DET
ejpam-5974	70	5	barbell	barbell	NOUN
ejpam-5974	70	6	graph	graph	NOUN
ejpam-5974	70	7	(	(	PUNCT
ejpam-5974	70	8	bn	bn	NOUN
ejpam-5974	70	9	)	)	PUNCT
ejpam-5974	70	10	where	where	SCONJ
ejpam-5974	70	11	n	n	PRON
ejpam-5974	70	12	≥	≥	NOUN
ejpam-5974	70	13	3	3	NUM
ejpam-5974	70	14	.	.	PUNCT
ejpam-5974	71	1	then	then	ADV
ejpam-5974	71	2	d(bn	d(bn	X
ejpam-5974	71	3	)	)	PUNCT
ejpam-5974	71	4	=	=	SYM
ejpam-5974	71	5	n2	n2	NOUN
ejpam-5974	71	6	−	−	PROPN
ejpam-5974	71	7	n+	n+	ADP
ejpam-5974	71	8	1	1	NUM
ejpam-5974	71	9	2n2	2n2	NUM
ejpam-5974	71	10	−	−	NOUN
ejpam-5974	71	11	n	n	NOUN
ejpam-5974	71	12	.	.	PUNCT
ejpam-5974	72	1	r.	r.	PROPN
ejpam-5974	72	2	sango	sango	PROPN
ejpam-5974	72	3	,	,	PUNCT
ejpam-5974	72	4	i.	i.	NOUN
ejpam-5974	72	5	cabahug	cabahug	PROPN
ejpam-5974	72	6	,	,	PUNCT
ejpam-5974	72	7	jr	jr	PROPN
ejpam-5974	72	8	/	/	SYM
ejpam-5974	72	9	eur	eur	PROPN
ejpam-5974	72	10	.	.	PUNCT
ejpam-5974	73	1	j.	j.	PROPN
ejpam-5974	73	2	pure	pure	PROPN
ejpam-5974	73	3	appl	appl	PROPN
ejpam-5974	73	4	.	.	PROPN
ejpam-5974	73	5	math	math	PROPN
ejpam-5974	73	6	,	,	PUNCT
ejpam-5974	73	7	18	18	NUM
ejpam-5974	73	8	(	(	PUNCT
ejpam-5974	73	9	3	3	NUM
ejpam-5974	73	10	)	)	PUNCT
ejpam-5974	73	11	(	(	PUNCT
ejpam-5974	73	12	2025	2025	NUM
ejpam-5974	73	13	)	)	PUNCT
ejpam-5974	73	14	,	,	PUNCT
ejpam-5974	73	15	5974	5974	NUM
ejpam-5974	73	16	4	4	NUM
ejpam-5974	73	17	of	of	ADP
ejpam-5974	73	18	16	16	NUM
ejpam-5974	73	19	8	8	NUM
ejpam-5974	73	20	4	4	NUM
ejpam-5974	73	21	2	2	NUM
ejpam-5974	73	22	1	1	NUM
ejpam-5974	73	23	3	3	NUM
ejpam-5974	73	24	5	5	NUM
ejpam-5974	73	25	96	96	NUM
ejpam-5974	73	26	7	7	NUM
ejpam-5974	73	27	10	10	NUM
ejpam-5974	73	28	11	11	NUM
ejpam-5974	73	29	figure	figure	NOUN
ejpam-5974	73	30	1	1	NUM
ejpam-5974	73	31	:	:	PUNCT
ejpam-5974	73	32	the	the	DET
ejpam-5974	73	33	density	density	NOUN
ejpam-5974	73	34	of	of	ADP
ejpam-5974	73	35	d(g	d(g	PROPN
ejpam-5974	73	36	)	)	PUNCT
ejpam-5974	73	37	=	=	PUNCT
ejpam-5974	74	1	0.218	0.218	NUM
ejpam-5974	74	2	proof	proof	NOUN
ejpam-5974	74	3	.	.	PUNCT
ejpam-5974	75	1	a	a	DET
ejpam-5974	75	2	barbell	barbell	NOUN
ejpam-5974	75	3	graph	graph	NOUN
ejpam-5974	75	4	bn	bn	NOUN
ejpam-5974	75	5	is	be	AUX
ejpam-5974	75	6	constructed	construct	VERB
ejpam-5974	75	7	by	by	ADP
ejpam-5974	75	8	taking	take	VERB
ejpam-5974	75	9	two	two	NUM
ejpam-5974	75	10	disjoint	disjoint	ADJ
ejpam-5974	75	11	complete	complete	ADJ
ejpam-5974	75	12	graph	graph	NOUN
ejpam-5974	75	13	kn	kn	PROPN
ejpam-5974	75	14	and	and	CCONJ
ejpam-5974	75	15	adding	add	VERB
ejpam-5974	75	16	exactly	exactly	ADV
ejpam-5974	75	17	one	one	NUM
ejpam-5974	75	18	edge	edge	NOUN
ejpam-5974	75	19	that	that	PRON
ejpam-5974	75	20	connects	connect	VERB
ejpam-5974	75	21	one	one	NUM
ejpam-5974	75	22	vertex	vertex	NOUN
ejpam-5974	75	23	first	first	ADV
ejpam-5974	75	24	in	in	ADP
ejpam-5974	75	25	kn1	kn1	NOUN
ejpam-5974	75	26	to	to	ADP
ejpam-5974	75	27	one	one	NUM
ejpam-5974	75	28	vertex	vertex	NOUN
ejpam-5974	75	29	in	in	ADP
ejpam-5974	75	30	the	the	DET
ejpam-5974	75	31	second	second	ADJ
ejpam-5974	75	32	kn2	kn2	NOUN
ejpam-5974	75	33	.	.	PUNCT
ejpam-5974	76	1	thus	thus	ADV
ejpam-5974	76	2	the	the	DET
ejpam-5974	76	3	number	number	NOUN
ejpam-5974	76	4	of	of	ADP
ejpam-5974	76	5	edges	edge	NOUN
ejpam-5974	76	6	of	of	ADP
ejpam-5974	76	7	the	the	DET
ejpam-5974	76	8	barbell	barbell	NOUN
ejpam-5974	76	9	graph	graph	NOUN
ejpam-5974	76	10	bn	bn	PROPN
ejpam-5974	76	11	is	be	AUX
ejpam-5974	76	12	:	:	PUNCT
ejpam-5974	76	13	|e(bn)|	|e(bn)|	NUM
ejpam-5974	76	14	=	=	SYM
ejpam-5974	76	15	2	2	NUM
ejpam-5974	76	16	(	(	PUNCT
ejpam-5974	76	17	n(n−	n(n−	NOUN
ejpam-5974	76	18	1	1	NUM
ejpam-5974	76	19	)	)	PUNCT
ejpam-5974	76	20	2	2	NUM
ejpam-5974	76	21	)	)	PUNCT
ejpam-5974	77	1	+	+	CCONJ
ejpam-5974	77	2	1	1	X
ejpam-5974	77	3	=	=	SYM
ejpam-5974	77	4	n(n−	n(n−	PRON
ejpam-5974	77	5	1	1	NUM
ejpam-5974	77	6	)	)	PUNCT
ejpam-5974	77	7	+	+	CCONJ
ejpam-5974	77	8	1	1	NUM
ejpam-5974	77	9	=	=	SYM
ejpam-5974	77	10	n2	n2	NOUN
ejpam-5974	77	11	−	−	PROPN
ejpam-5974	77	12	n+	n+	ADP
ejpam-5974	77	13	1	1	NUM
ejpam-5974	77	14	.	.	PUNCT
ejpam-5974	78	1	and	and	CCONJ
ejpam-5974	78	2	the	the	DET
ejpam-5974	78	3	number	number	NOUN
ejpam-5974	78	4	of	of	ADP
ejpam-5974	78	5	vertices	vertex	NOUN
ejpam-5974	78	6	of	of	ADP
ejpam-5974	78	7	the	the	DET
ejpam-5974	78	8	barbell	barbell	NOUN
ejpam-5974	78	9	graph	graph	NOUN
ejpam-5974	78	10	bn	bn	PROPN
ejpam-5974	78	11	is	be	AUX
ejpam-5974	78	12	|v	|v	X
ejpam-5974	78	13	(	(	PUNCT
ejpam-5974	78	14	bn)|	bn)|	X
ejpam-5974	78	15	=	=	SYM
ejpam-5974	78	16	2n	2n	NUM
ejpam-5974	78	17	.	.	PUNCT
ejpam-5974	79	1	therefore	therefore	ADV
ejpam-5974	79	2	,	,	PUNCT
ejpam-5974	79	3	g(bn	g(bn	X
ejpam-5974	79	4	)	)	PUNCT
ejpam-5974	79	5	=	=	SYM
ejpam-5974	79	6	2|e(bn)|	2|e(bn)|	PROPN
ejpam-5974	79	7	|v	|v	X
ejpam-5974	79	8	(	(	PUNCT
ejpam-5974	79	9	bn)|(|v	bn)|(|v	X
ejpam-5974	79	10	(	(	PUNCT
ejpam-5974	79	11	bn)|	bn)|	PROPN
ejpam-5974	79	12	−	−	ADP
ejpam-5974	79	13	1	1	NUM
ejpam-5974	79	14	)	)	PUNCT
ejpam-5974	79	15	=	=	SYM
ejpam-5974	79	16	2(n2	2(n2	NUM
ejpam-5974	79	17	−	−	NOUN
ejpam-5974	79	18	n+	n+	NOUN
ejpam-5974	79	19	1	1	NUM
ejpam-5974	79	20	)	)	PUNCT
ejpam-5974	79	21	2n(2n−	2n(2n−	NUM
ejpam-5974	79	22	1	1	NUM
ejpam-5974	79	23	)	)	PUNCT
ejpam-5974	79	24	=	=	SYM
ejpam-5974	79	25	n2	n2	NOUN
ejpam-5974	79	26	−	−	PROPN
ejpam-5974	79	27	n+	n+	ADP
ejpam-5974	79	28	1	1	NUM
ejpam-5974	79	29	2n2	2n2	NUM
ejpam-5974	79	30	−	−	NOUN
ejpam-5974	79	31	n	n	NOUN
ejpam-5974	79	32	.	.	PUNCT
ejpam-5974	80	1	illustration	illustration	NOUN
ejpam-5974	80	2	:	:	PUNCT
ejpam-5974	80	3	consider	consider	VERB
ejpam-5974	80	4	figure	figure	NOUN
ejpam-5974	80	5	2	2	NUM
ejpam-5974	80	6	,	,	PUNCT
ejpam-5974	80	7	it	it	PRON
ejpam-5974	80	8	shows	show	VERB
ejpam-5974	80	9	that	that	SCONJ
ejpam-5974	80	10	the	the	DET
ejpam-5974	80	11	number	number	NOUN
ejpam-5974	80	12	of	of	ADP
ejpam-5974	80	13	edges	edge	NOUN
ejpam-5974	80	14	and	and	CCONJ
ejpam-5974	80	15	vertices	vertex	NOUN
ejpam-5974	80	16	of	of	ADP
ejpam-5974	80	17	b4	b4	NOUN
ejpam-5974	80	18	are	be	AUX
ejpam-5974	80	19	|e|	|e|	PRON
ejpam-5974	80	20	=	=	SYM
ejpam-5974	80	21	13	13	NUM
ejpam-5974	80	22	and	and	CCONJ
ejpam-5974	80	23	|v	|v	PROPN
ejpam-5974	80	24	|	|	ADV
ejpam-5974	80	25	=	=	NOUN
ejpam-5974	80	26	8	8	NUM
ejpam-5974	80	27	respectively	respectively	ADV
ejpam-5974	80	28	.	.	PUNCT
ejpam-5974	81	1	clearly	clearly	ADV
ejpam-5974	81	2	,	,	PUNCT
ejpam-5974	81	3	d(b5	d(b5	ADJ
ejpam-5974	81	4	)	)	PUNCT
ejpam-5974	81	5	=	=	SYM
ejpam-5974	81	6	2|e(b5)|	2|e(b5)|	NUM
ejpam-5974	81	7	|v	|v	X
ejpam-5974	81	8	(	(	PUNCT
ejpam-5974	81	9	b5)|(|v	b5)|(|v	PROPN
ejpam-5974	81	10	(	(	PUNCT
ejpam-5974	81	11	b5)|	b5)|	PROPN
ejpam-5974	81	12	−	−	PROPN
ejpam-5974	81	13	1	1	NUM
ejpam-5974	81	14	)	)	PUNCT
ejpam-5974	81	15	2|13|	2|13|	NUM
ejpam-5974	81	16	8(8−	8(8−	NUM
ejpam-5974	81	17	1	1	NUM
ejpam-5974	81	18	)	)	PUNCT
ejpam-5974	81	19	=	=	NOUN
ejpam-5974	81	20	26	26	NUM
ejpam-5974	81	21	8(7	8(7	NUM
ejpam-5974	81	22	)	)	PUNCT
ejpam-5974	81	23	=	=	PUNCT
ejpam-5974	82	1	26	26	NUM
ejpam-5974	82	2	56	56	NUM
ejpam-5974	82	3	=	=	SYM
ejpam-5974	82	4	0.4642	0.4642	NUM
ejpam-5974	82	5	or	or	CCONJ
ejpam-5974	82	6	d(b5	d(b5	ADJ
ejpam-5974	82	7	)	)	PUNCT
ejpam-5974	83	1	=	=	SYM
ejpam-5974	83	2	n2	n2	NOUN
ejpam-5974	83	3	−	−	PROPN
ejpam-5974	83	4	n+	n+	ADP
ejpam-5974	83	5	1	1	NUM
ejpam-5974	83	6	2n2	2n2	NUM
ejpam-5974	83	7	−	−	NOUN
ejpam-5974	83	8	n	n	NOUN
ejpam-5974	83	9	=	=	SYM
ejpam-5974	83	10	2(42	2(42	NUM
ejpam-5974	83	11	−	−	PROPN
ejpam-5974	83	12	2(4	2(4	NUM
ejpam-5974	83	13	)	)	PUNCT
ejpam-5974	84	1	+	+	CCONJ
ejpam-5974	84	2	2	2	X
ejpam-5974	84	3	)	)	PUNCT
ejpam-5974	84	4	4(42)−	4(42)−	NUM
ejpam-5974	84	5	2(4	2(4	NUM
ejpam-5974	84	6	)	)	PUNCT
ejpam-5974	84	7	=	=	PUNCT
ejpam-5974	85	1	26	26	NUM
ejpam-5974	85	2	56	56	NUM
ejpam-5974	85	3	=	=	SYM
ejpam-5974	85	4	0.4642	0.4642	NUM
ejpam-5974	85	5	.	.	PUNCT
ejpam-5974	86	1	figure	figure	NOUN
ejpam-5974	86	2	2	2	NUM
ejpam-5974	86	3	:	:	PUNCT
ejpam-5974	86	4	the	the	DET
ejpam-5974	86	5	barbell	barbell	NOUN
ejpam-5974	86	6	graph	graph	NOUN
ejpam-5974	86	7	b4	b4	NOUN
ejpam-5974	86	8	theorem	theorem	VERB
ejpam-5974	86	9	2	2	NUM
ejpam-5974	86	10	.	.	PUNCT
ejpam-5974	87	1	let	let	VERB
ejpam-5974	87	2	g	g	PRON
ejpam-5974	87	3	be	be	AUX
ejpam-5974	87	4	a	a	DET
ejpam-5974	87	5	friendship	friendship	NOUN
ejpam-5974	87	6	graph	graph	NOUN
ejpam-5974	87	7	(	(	PUNCT
ejpam-5974	87	8	frn	frn	PROPN
ejpam-5974	87	9	)	)	PUNCT
ejpam-5974	87	10	where	where	SCONJ
ejpam-5974	87	11	n	n	PRON
ejpam-5974	87	12	≥	≥	NOUN
ejpam-5974	87	13	2	2	NUM
ejpam-5974	87	14	.	.	PUNCT
ejpam-5974	88	1	then	then	ADV
ejpam-5974	88	2	d(frn	d(frn	PROPN
ejpam-5974	88	3	)	)	PUNCT
ejpam-5974	89	1	=	=	NOUN
ejpam-5974	89	2	3n	3n	NUM
ejpam-5974	89	3	2n2	2n2	NUM
ejpam-5974	89	4	+	+	CCONJ
ejpam-5974	89	5	n	n	X
ejpam-5974	89	6	.	.	PUNCT
ejpam-5974	90	1	r.	r.	PROPN
ejpam-5974	90	2	sango	sango	PROPN
ejpam-5974	90	3	,	,	PUNCT
ejpam-5974	90	4	i.	i.	NOUN
ejpam-5974	90	5	cabahug	cabahug	PROPN
ejpam-5974	90	6	,	,	PUNCT
ejpam-5974	90	7	jr	jr	PROPN
ejpam-5974	90	8	/	/	SYM
ejpam-5974	90	9	eur	eur	PROPN
ejpam-5974	90	10	.	.	PUNCT
ejpam-5974	91	1	j.	j.	PROPN
ejpam-5974	91	2	pure	pure	PROPN
ejpam-5974	91	3	appl	appl	PROPN
ejpam-5974	91	4	.	.	PROPN
ejpam-5974	91	5	math	math	PROPN
ejpam-5974	91	6	,	,	PUNCT
ejpam-5974	91	7	18	18	NUM
ejpam-5974	91	8	(	(	PUNCT
ejpam-5974	91	9	3	3	NUM
ejpam-5974	91	10	)	)	PUNCT
ejpam-5974	91	11	(	(	PUNCT
ejpam-5974	91	12	2025	2025	NUM
ejpam-5974	91	13	)	)	PUNCT
ejpam-5974	91	14	,	,	PUNCT
ejpam-5974	91	15	5974	5974	NUM
ejpam-5974	91	16	5	5	NUM
ejpam-5974	91	17	of	of	ADP
ejpam-5974	91	18	16	16	NUM
ejpam-5974	91	19	proof	proof	NOUN
ejpam-5974	91	20	.	.	PUNCT
ejpam-5974	92	1	a	a	DET
ejpam-5974	92	2	friendship	friendship	NOUN
ejpam-5974	92	3	graph	graph	NOUN
ejpam-5974	92	4	frn	frn	PROPN
ejpam-5974	92	5	is	be	AUX
ejpam-5974	92	6	constructed	construct	VERB
ejpam-5974	92	7	by	by	ADP
ejpam-5974	92	8	n	n	NOUN
ejpam-5974	92	9	triangles	triangle	NOUN
ejpam-5974	92	10	with	with	ADP
ejpam-5974	92	11	a	a	DET
ejpam-5974	92	12	common	common	ADJ
ejpam-5974	92	13	vertex	vertex	NOUN
ejpam-5974	92	14	.	.	PUNCT
ejpam-5974	93	1	thus	thus	ADV
ejpam-5974	93	2	the	the	DET
ejpam-5974	93	3	number	number	NOUN
ejpam-5974	93	4	of	of	ADP
ejpam-5974	93	5	edges	edge	NOUN
ejpam-5974	93	6	and	and	CCONJ
ejpam-5974	93	7	vertices	vertex	NOUN
ejpam-5974	93	8	of	of	ADP
ejpam-5974	93	9	frn	frn	PROPN
ejpam-5974	93	10	are	be	AUX
ejpam-5974	93	11	|e(frn)|	|e(frn)|	ADP
ejpam-5974	93	12	=	=	SYM
ejpam-5974	93	13	3n	3n	NOUN
ejpam-5974	93	14	and	and	CCONJ
ejpam-5974	93	15	|v	|v	PROPN
ejpam-5974	93	16	(	(	PUNCT
ejpam-5974	93	17	frn)|	frn)|	X
ejpam-5974	93	18	=	=	SYM
ejpam-5974	93	19	3n−	3n−	PROPN
ejpam-5974	93	20	2	2	NUM
ejpam-5974	93	21	respectively	respectively	ADV
ejpam-5974	93	22	.	.	PUNCT
ejpam-5974	94	1	therefore	therefore	ADV
ejpam-5974	94	2	,	,	PUNCT
ejpam-5974	94	3	d(frn	d(frn	PROPN
ejpam-5974	94	4	)	)	PUNCT
ejpam-5974	95	1	=	=	PUNCT
ejpam-5974	96	1	2|e(frn)|	2|e(frn)|	NUM
ejpam-5974	96	2	|v	|v	X
ejpam-5974	96	3	(	(	PUNCT
ejpam-5974	96	4	frn)|(|v	frn)|(|v	PROPN
ejpam-5974	96	5	(	(	PUNCT
ejpam-5974	96	6	frn)|	frn)|	NOUN
ejpam-5974	96	7	−	−	PROPN
ejpam-5974	96	8	1	1	NUM
ejpam-5974	96	9	)	)	PUNCT
ejpam-5974	96	10	=	=	SYM
ejpam-5974	96	11	2(3n	2(3n	NUM
ejpam-5974	96	12	)	)	PUNCT
ejpam-5974	96	13	(	(	PUNCT
ejpam-5974	96	14	2n+	2n+	NUM
ejpam-5974	96	15	1)(2n+	1)(2n+	NUM
ejpam-5974	96	16	1−	1−	NUM
ejpam-5974	96	17	1	1	NUM
ejpam-5974	96	18	)	)	PUNCT
ejpam-5974	96	19	=	=	SYM
ejpam-5974	96	20	6n	6n	NOUN
ejpam-5974	96	21	(	(	PUNCT
ejpam-5974	96	22	2n+	2n+	NUM
ejpam-5974	96	23	1)(2n	1)(2n	NUM
ejpam-5974	96	24	)	)	PUNCT
ejpam-5974	96	25	=	=	SYM
ejpam-5974	96	26	3n	3n	NOUN
ejpam-5974	96	27	(	(	PUNCT
ejpam-5974	96	28	2n+	2n+	NUM
ejpam-5974	96	29	1)n	1)n	NOUN
ejpam-5974	96	30	=	=	NOUN
ejpam-5974	96	31	3n	3n	NUM
ejpam-5974	96	32	2n2	2n2	NUM
ejpam-5974	96	33	+	+	CCONJ
ejpam-5974	96	34	n	n	NOUN
ejpam-5974	96	35	.	.	PUNCT
ejpam-5974	96	36	illustration	illustration	NOUN
ejpam-5974	96	37	:	:	PUNCT
ejpam-5974	96	38	consider	consider	VERB
ejpam-5974	96	39	figure	figure	NOUN
ejpam-5974	96	40	3	3	NUM
ejpam-5974	96	41	,	,	PUNCT
ejpam-5974	96	42	it	it	PRON
ejpam-5974	96	43	shows	show	VERB
ejpam-5974	96	44	that	that	SCONJ
ejpam-5974	96	45	the	the	DET
ejpam-5974	96	46	number	number	NOUN
ejpam-5974	96	47	of	of	ADP
ejpam-5974	96	48	edges	edge	NOUN
ejpam-5974	96	49	and	and	CCONJ
ejpam-5974	96	50	vertices	vertex	NOUN
ejpam-5974	96	51	of	of	ADP
ejpam-5974	96	52	frn	frn	PROPN
ejpam-5974	96	53	are	be	AUX
ejpam-5974	96	54	|e(fr3)|	|e(fr3)|	PUNCT
ejpam-5974	96	55	=	=	SYM
ejpam-5974	96	56	9	9	NUM
ejpam-5974	96	57	and	and	CCONJ
ejpam-5974	96	58	|v	|v	PROPN
ejpam-5974	96	59	(	(	PUNCT
ejpam-5974	96	60	fr3)|	fr3)|	NOUN
ejpam-5974	96	61	=	=	NOUN
ejpam-5974	96	62	7	7	NUM
ejpam-5974	96	63	respectively	respectively	ADV
ejpam-5974	96	64	.	.	PUNCT
ejpam-5974	97	1	clearly	clearly	ADV
ejpam-5974	97	2	,	,	PUNCT
ejpam-5974	97	3	d(f3	d(f3	NOUN
ejpam-5974	97	4	)	)	PUNCT
ejpam-5974	97	5	=	=	PUNCT
ejpam-5974	98	1	2|e(fr3)|	2|e(fr3)|	NUM
ejpam-5974	98	2	|v	|v	NOUN
ejpam-5974	98	3	(	(	PUNCT
ejpam-5974	98	4	fr3)|(|v	fr3)|(|v	PROPN
ejpam-5974	98	5	(	(	PUNCT
ejpam-5974	98	6	fr3)|	fr3)|	NOUN
ejpam-5974	98	7	−	−	NOUN
ejpam-5974	98	8	1	1	NUM
ejpam-5974	98	9	)	)	PUNCT
ejpam-5974	98	10	=	=	NOUN
ejpam-5974	98	11	18	18	NUM
ejpam-5974	98	12	7(6	7(6	NUM
ejpam-5974	98	13	)	)	PUNCT
ejpam-5974	98	14	=	=	SYM
ejpam-5974	98	15	18	18	NUM
ejpam-5974	98	16	42	42	NUM
ejpam-5974	98	17	=	=	SYM
ejpam-5974	98	18	0.429	0.429	NUM
ejpam-5974	98	19	or	or	CCONJ
ejpam-5974	98	20	d(f3	d(f3	NOUN
ejpam-5974	98	21	)	)	PUNCT
ejpam-5974	98	22	=	=	PUNCT
ejpam-5974	98	23	3n	3n	NOUN
ejpam-5974	98	24	2n2	2n2	NUM
ejpam-5974	98	25	+	+	CCONJ
ejpam-5974	98	26	n	n	CCONJ
ejpam-5974	98	27	=	=	SYM
ejpam-5974	98	28	3(3	3(3	NUM
ejpam-5974	98	29	)	)	PUNCT
ejpam-5974	98	30	3(3)2	3(3)2	NOUN
ejpam-5974	98	31	−	−	PROPN
ejpam-5974	98	32	5(3	5(3	NUM
ejpam-5974	98	33	)	)	PUNCT
ejpam-5974	98	34	+	+	CCONJ
ejpam-5974	98	35	2	2	NUM
ejpam-5974	98	36	=	=	SYM
ejpam-5974	98	37	9	9	NUM
ejpam-5974	98	38	9−	9−	NUM
ejpam-5974	98	39	15	15	NUM
ejpam-5974	98	40	+	+	CCONJ
ejpam-5974	98	41	2	2	NUM
ejpam-5974	98	42	=	=	SYM
ejpam-5974	98	43	0.429	0.429	NUM
ejpam-5974	98	44	.	.	PUNCT
ejpam-5974	99	1	v1	v1	PROPN
ejpam-5974	99	2	v2	v2	PROPN
ejpam-5974	99	3	v3	v3	PROPN
ejpam-5974	99	4	v4	v4	PROPN
ejpam-5974	99	5	v5v6	v5v6	PROPN
ejpam-5974	99	6	v0	v0	NOUN
ejpam-5974	99	7	figure	figure	NOUN
ejpam-5974	99	8	3	3	NUM
ejpam-5974	99	9	:	:	PUNCT
ejpam-5974	99	10	the	the	DET
ejpam-5974	99	11	friendship	friendship	NOUN
ejpam-5974	99	12	graph	graph	NOUN
ejpam-5974	99	13	fr3	fr3	PROPN
ejpam-5974	99	14	theorem	theorem	NOUN
ejpam-5974	99	15	3	3	X
ejpam-5974	99	16	.	.	PUNCT
ejpam-5974	100	1	let	let	VERB
ejpam-5974	100	2	g	g	PRON
ejpam-5974	100	3	be	be	AUX
ejpam-5974	100	4	a	a	DET
ejpam-5974	100	5	sunlet	sunlet	NOUN
ejpam-5974	100	6	graph	graph	NOUN
ejpam-5974	100	7	(	(	PUNCT
ejpam-5974	100	8	sn	sn	PROPN
ejpam-5974	100	9	)	)	PUNCT
ejpam-5974	100	10	where	where	SCONJ
ejpam-5974	100	11	n	n	PRON
ejpam-5974	100	12	≥	≥	NOUN
ejpam-5974	100	13	3	3	NUM
ejpam-5974	100	14	.	.	PUNCT
ejpam-5974	101	1	then	then	ADV
ejpam-5974	101	2	d(sn	d(sn	ADJ
ejpam-5974	101	3	)	)	PUNCT
ejpam-5974	101	4	=	=	SYM
ejpam-5974	101	5	2	2	NUM
ejpam-5974	101	6	2n−	2n−	NUM
ejpam-5974	101	7	1	1	NUM
ejpam-5974	101	8	.	.	PUNCT
ejpam-5974	102	1	proof	proof	NOUN
ejpam-5974	102	2	.	.	PUNCT
ejpam-5974	103	1	a	a	DET
ejpam-5974	103	2	sunlet	sunlet	NOUN
ejpam-5974	103	3	graph	graph	NOUN
ejpam-5974	103	4	sn	sn	PROPN
ejpam-5974	103	5	is	be	AUX
ejpam-5974	103	6	constructed	construct	VERB
ejpam-5974	103	7	by	by	ADP
ejpam-5974	103	8	adding	add	VERB
ejpam-5974	103	9	n	n	DET
ejpam-5974	103	10	pendant	pendant	ADJ
ejpam-5974	103	11	edges	edge	NOUN
ejpam-5974	103	12	to	to	ADP
ejpam-5974	103	13	a	a	DET
ejpam-5974	103	14	cycle	cycle	NOUN
ejpam-5974	103	15	graph	graph	NOUN
ejpam-5974	103	16	cn	cn	PROPN
ejpam-5974	103	17	.	.	PUNCT
ejpam-5974	104	1	thus	thus	ADV
ejpam-5974	104	2	,	,	PUNCT
ejpam-5974	104	3	the	the	DET
ejpam-5974	104	4	number	number	NOUN
ejpam-5974	104	5	of	of	ADP
ejpam-5974	104	6	edges	edge	NOUN
ejpam-5974	104	7	and	and	CCONJ
ejpam-5974	104	8	vertices	vertex	NOUN
ejpam-5974	104	9	of	of	ADP
ejpam-5974	104	10	the	the	DET
ejpam-5974	104	11	sunlet	sunlet	NOUN
ejpam-5974	104	12	graph	graph	NOUN
ejpam-5974	104	13	sn	sn	PROPN
ejpam-5974	104	14	is	be	AUX
ejpam-5974	104	15	|e(sn)|	|e(sn)|	ADV
ejpam-5974	104	16	=	=	SYM
ejpam-5974	104	17	|v	|v	X
ejpam-5974	104	18	(	(	PUNCT
ejpam-5974	104	19	sn)|	sn)|	NOUN
ejpam-5974	104	20	=	=	SYM
ejpam-5974	104	21	2n	2n	NUM
ejpam-5974	104	22	.	.	PUNCT
ejpam-5974	105	1	therefore	therefore	ADV
ejpam-5974	105	2	,	,	PUNCT
ejpam-5974	105	3	d(sn	d(sn	NOUN
ejpam-5974	105	4	)	)	PUNCT
ejpam-5974	105	5	=	=	SYM
ejpam-5974	106	1	2|e(sn)|	2|e(sn)|	NUM
ejpam-5974	106	2	|v	|v	NOUN
ejpam-5974	106	3	(	(	PUNCT
ejpam-5974	106	4	sn)|(|v	sn)|(|v	X
ejpam-5974	106	5	(	(	PUNCT
ejpam-5974	106	6	sn)|	sn)|	VERB
ejpam-5974	106	7	−	−	NOUN
ejpam-5974	106	8	1	1	NUM
ejpam-5974	106	9	)	)	PUNCT
ejpam-5974	106	10	=	=	SYM
ejpam-5974	106	11	2(2n	2(2n	NUM
ejpam-5974	106	12	)	)	PUNCT
ejpam-5974	106	13	2n(2n−	2n(2n−	NUM
ejpam-5974	106	14	1	1	NUM
ejpam-5974	106	15	)	)	PUNCT
ejpam-5974	106	16	=	=	SYM
ejpam-5974	106	17	2	2	NUM
ejpam-5974	106	18	2n−	2n−	NUM
ejpam-5974	106	19	1	1	NUM
ejpam-5974	106	20	.	.	PUNCT
ejpam-5974	106	21	r.	r.	PROPN
ejpam-5974	106	22	sango	sango	PROPN
ejpam-5974	106	23	,	,	PUNCT
ejpam-5974	106	24	i.	i.	NOUN
ejpam-5974	106	25	cabahug	cabahug	PROPN
ejpam-5974	106	26	,	,	PUNCT
ejpam-5974	106	27	jr	jr	PROPN
ejpam-5974	106	28	/	/	SYM
ejpam-5974	106	29	eur	eur	PROPN
ejpam-5974	106	30	.	.	PUNCT
ejpam-5974	107	1	j.	j.	PROPN
ejpam-5974	107	2	pure	pure	PROPN
ejpam-5974	107	3	appl	appl	PROPN
ejpam-5974	107	4	.	.	PROPN
ejpam-5974	107	5	math	math	PROPN
ejpam-5974	107	6	,	,	PUNCT
ejpam-5974	107	7	18	18	NUM
ejpam-5974	107	8	(	(	PUNCT
ejpam-5974	107	9	3	3	NUM
ejpam-5974	107	10	)	)	PUNCT
ejpam-5974	107	11	(	(	PUNCT
ejpam-5974	107	12	2025	2025	NUM
ejpam-5974	107	13	)	)	PUNCT
ejpam-5974	107	14	,	,	PUNCT
ejpam-5974	107	15	5974	5974	NUM
ejpam-5974	107	16	6	6	NUM
ejpam-5974	107	17	of	of	ADP
ejpam-5974	107	18	16	16	NUM
ejpam-5974	107	19	illustration	illustration	NOUN
ejpam-5974	107	20	:	:	PUNCT
ejpam-5974	107	21	consider	consider	VERB
ejpam-5974	107	22	figure	figure	NOUN
ejpam-5974	107	23	4	4	NUM
ejpam-5974	107	24	,	,	PUNCT
ejpam-5974	107	25	it	it	PRON
ejpam-5974	107	26	shows	show	VERB
ejpam-5974	107	27	that	that	SCONJ
ejpam-5974	107	28	the	the	DET
ejpam-5974	107	29	number	number	NOUN
ejpam-5974	107	30	of	of	ADP
ejpam-5974	107	31	edges	edge	NOUN
ejpam-5974	107	32	and	and	CCONJ
ejpam-5974	107	33	vertices	vertex	NOUN
ejpam-5974	107	34	of	of	ADP
ejpam-5974	107	35	s4	s4	PROPN
ejpam-5974	107	36	are	be	AUX
ejpam-5974	107	37	|e(s4)|	|e(s4)|	ADP
ejpam-5974	107	38	=	=	SYM
ejpam-5974	107	39	8	8	NUM
ejpam-5974	107	40	and	and	CCONJ
ejpam-5974	107	41	|v	|v	PROPN
ejpam-5974	107	42	(	(	PUNCT
ejpam-5974	107	43	s4)|	s4)|	PROPN
ejpam-5974	107	44	=	=	PROPN
ejpam-5974	107	45	8	8	NUM
ejpam-5974	107	46	respectively	respectively	ADV
ejpam-5974	107	47	.	.	PUNCT
ejpam-5974	108	1	clearly	clearly	ADV
ejpam-5974	108	2	,	,	PUNCT
ejpam-5974	108	3	d(s4	d(s4	X
ejpam-5974	108	4	)	)	PUNCT
ejpam-5974	109	1	=	=	SYM
ejpam-5974	109	2	2|e(s4)|	2|e(s4)|	NUM
ejpam-5974	109	3	|v	|v	NOUN
ejpam-5974	109	4	(	(	PUNCT
ejpam-5974	109	5	s4)|(|v	s4)|(|v	PROPN
ejpam-5974	109	6	(	(	PUNCT
ejpam-5974	109	7	s4)|	s4)|	PROPN
ejpam-5974	109	8	−	−	PROPN
ejpam-5974	109	9	1	1	NUM
ejpam-5974	109	10	)	)	PUNCT
ejpam-5974	109	11	=	=	SYM
ejpam-5974	109	12	2|8|	2|8|	NUM
ejpam-5974	109	13	8(7	8(7	NUM
ejpam-5974	109	14	)	)	PUNCT
ejpam-5974	109	15	=	=	PUNCT
ejpam-5974	110	1	16	16	NUM
ejpam-5974	110	2	56	56	NUM
ejpam-5974	110	3	=	=	SYM
ejpam-5974	110	4	2	2	NUM
ejpam-5974	110	5	7	7	NUM
ejpam-5974	110	6	=	=	SYM
ejpam-5974	110	7	0.2857	0.2857	NUM
ejpam-5974	110	8	or	or	CCONJ
ejpam-5974	110	9	d(s4	d(s4	NOUN
ejpam-5974	110	10	)	)	PUNCT
ejpam-5974	110	11	=	=	SYM
ejpam-5974	110	12	2	2	NUM
ejpam-5974	110	13	2n−	2n−	NUM
ejpam-5974	110	14	1	1	NUM
ejpam-5974	110	15	=	=	SYM
ejpam-5974	110	16	2	2	NUM
ejpam-5974	110	17	2(4	2(4	NUM
ejpam-5974	110	18	)	)	PUNCT
ejpam-5974	111	1	+	+	CCONJ
ejpam-5974	111	2	1	1	NUM
ejpam-5974	111	3	=	=	SYM
ejpam-5974	111	4	2	2	NUM
ejpam-5974	111	5	7	7	NUM
ejpam-5974	111	6	=	=	SYM
ejpam-5974	111	7	0.2857	0.2857	NUM
ejpam-5974	111	8	.	.	PUNCT
ejpam-5974	112	1	v1	v1	PROPN
ejpam-5974	112	2	v2	v2	PROPN
ejpam-5974	112	3	v3v4	v3v4	X
ejpam-5974	112	4	v′1	v′1	PROPN
ejpam-5974	112	5	v′2	v′2	PROPN
ejpam-5974	112	6	v′4	v′4	PROPN
ejpam-5974	112	7	v′3	v′3	NOUN
ejpam-5974	112	8	figure	figure	NOUN
ejpam-5974	112	9	4	4	NUM
ejpam-5974	112	10	:	:	PUNCT
ejpam-5974	112	11	the	the	DET
ejpam-5974	112	12	sunlet	sunlet	NOUN
ejpam-5974	112	13	graph	graph	NOUN
ejpam-5974	112	14	s4	s4	PROPN
ejpam-5974	112	15	theorem	theorem	ADJ
ejpam-5974	112	16	4	4	NUM
ejpam-5974	112	17	.	.	PUNCT
ejpam-5974	113	1	let	let	VERB
ejpam-5974	113	2	g	g	PRON
ejpam-5974	113	3	be	be	AUX
ejpam-5974	113	4	a	a	DET
ejpam-5974	113	5	banana	banana	NOUN
ejpam-5974	113	6	graph	graph	NOUN
ejpam-5974	113	7	bn	bn	PROPN
ejpam-5974	113	8	,	,	PUNCT
ejpam-5974	113	9	k	k	PROPN
ejpam-5974	113	10	where	where	SCONJ
ejpam-5974	113	11	n	n	PRON
ejpam-5974	113	12	≥	≥	X
ejpam-5974	113	13	2	2	NUM
ejpam-5974	113	14	and	and	CCONJ
ejpam-5974	113	15	k	k	PROPN
ejpam-5974	113	16	≥	≥	NUM
ejpam-5974	113	17	4	4	NUM
ejpam-5974	113	18	.	.	PUNCT
ejpam-5974	114	1	then	then	ADV
ejpam-5974	114	2	d(bn	d(bn	PROPN
ejpam-5974	114	3	,	,	PUNCT
ejpam-5974	114	4	k	k	NOUN
ejpam-5974	114	5	)	)	PUNCT
ejpam-5974	114	6	=	=	SYM
ejpam-5974	114	7	2	2	NUM
ejpam-5974	114	8	nk	nk	NOUN
ejpam-5974	114	9	+	+	NOUN
ejpam-5974	114	10	1	1	NUM
ejpam-5974	114	11	.	.	PUNCT
ejpam-5974	115	1	proof	proof	NOUN
ejpam-5974	115	2	.	.	PUNCT
ejpam-5974	116	1	a	a	DET
ejpam-5974	116	2	banana	banana	NOUN
ejpam-5974	116	3	graph	graph	NOUN
ejpam-5974	116	4	bn	bn	PROPN
ejpam-5974	116	5	,	,	PUNCT
ejpam-5974	116	6	k	k	PROPN
ejpam-5974	116	7	is	be	AUX
ejpam-5974	116	8	constructed	construct	VERB
ejpam-5974	116	9	by	by	ADP
ejpam-5974	116	10	connecting	connect	VERB
ejpam-5974	116	11	one	one	NUM
ejpam-5974	116	12	leaf	leaf	NOUN
ejpam-5974	116	13	of	of	ADP
ejpam-5974	116	14	each	each	DET
ejpam-5974	116	15	n	n	PROPN
ejpam-5974	116	16	copies	copy	NOUN
ejpam-5974	116	17	of	of	ADP
ejpam-5974	116	18	k	k	PROPN
ejpam-5974	116	19	-	-	PUNCT
ejpam-5974	116	20	star	star	NOUN
ejpam-5974	116	21	graph	graph	NOUN
ejpam-5974	116	22	with	with	ADP
ejpam-5974	116	23	one	one	NUM
ejpam-5974	116	24	root	root	NOUN
ejpam-5974	116	25	vertex	vertex	NOUN
ejpam-5974	116	26	.	.	PUNCT
ejpam-5974	117	1	thus	thus	ADV
ejpam-5974	117	2	,	,	PUNCT
ejpam-5974	117	3	the	the	DET
ejpam-5974	117	4	number	number	NOUN
ejpam-5974	117	5	of	of	ADP
ejpam-5974	117	6	edges	edge	NOUN
ejpam-5974	117	7	and	and	CCONJ
ejpam-5974	117	8	vertices	vertex	NOUN
ejpam-5974	117	9	of	of	ADP
ejpam-5974	117	10	the	the	DET
ejpam-5974	117	11	banana	banana	NOUN
ejpam-5974	117	12	graph	graph	NOUN
ejpam-5974	117	13	bn	bn	PROPN
ejpam-5974	117	14	,	,	PUNCT
ejpam-5974	117	15	k	k	PROPN
ejpam-5974	117	16	are	be	AUX
ejpam-5974	117	17	|e(bn	|e(bn	PROPN
ejpam-5974	117	18	,	,	PUNCT
ejpam-5974	117	19	k)|	k)|	NOUN
ejpam-5974	117	20	=	=	SYM
ejpam-5974	117	21	nk	nk	PROPN
ejpam-5974	117	22	and	and	CCONJ
ejpam-5974	117	23	|v	|v	PROPN
ejpam-5974	117	24	(	(	PUNCT
ejpam-5974	117	25	bn	bn	INTJ
ejpam-5974	117	26	,	,	PUNCT
ejpam-5974	117	27	k)|	k)|	NOUN
ejpam-5974	117	28	=	=	PUNCT
ejpam-5974	117	29	nk	nk	PROPN
ejpam-5974	118	1	+	+	NOUN
ejpam-5974	118	2	1	1	NUM
ejpam-5974	118	3	respectively	respectively	ADV
ejpam-5974	118	4	.	.	PUNCT
ejpam-5974	119	1	therefore	therefore	ADV
ejpam-5974	119	2	,	,	PUNCT
ejpam-5974	119	3	d(bn	d(bn	PROPN
ejpam-5974	119	4	,	,	PUNCT
ejpam-5974	119	5	k	k	NOUN
ejpam-5974	119	6	)	)	PUNCT
ejpam-5974	119	7	=	=	SYM
ejpam-5974	119	8	2|e(bn	2|e(bn	PROPN
ejpam-5974	119	9	,	,	PUNCT
ejpam-5974	119	10	k)|	k)|	PROPN
ejpam-5974	119	11	|v	|v	PROPN
ejpam-5974	119	12	(	(	PUNCT
ejpam-5974	119	13	bn	bn	INTJ
ejpam-5974	119	14	,	,	PUNCT
ejpam-5974	119	15	k)|(|v	k)|(|v	X
ejpam-5974	119	16	(	(	PUNCT
ejpam-5974	119	17	bn	bn	X
ejpam-5974	119	18	,	,	PUNCT
ejpam-5974	119	19	k)|	k)|	NOUN
ejpam-5974	119	20	−	−	NOUN
ejpam-5974	119	21	1	1	NUM
ejpam-5974	119	22	)	)	PUNCT
ejpam-5974	119	23	=	=	SYM
ejpam-5974	119	24	2(nk	2(nk	X
ejpam-5974	119	25	)	)	PUNCT
ejpam-5974	119	26	nk	nk	PROPN
ejpam-5974	120	1	+	+	CCONJ
ejpam-5974	120	2	1(nk	1(nk	NUM
ejpam-5974	120	3	+	+	CCONJ
ejpam-5974	120	4	1−	1−	NUM
ejpam-5974	120	5	1	1	NUM
ejpam-5974	120	6	)	)	PUNCT
ejpam-5974	121	1	=	=	SYM
ejpam-5974	121	2	2nk	2nk	NOUN
ejpam-5974	121	3	(	(	PUNCT
ejpam-5974	121	4	nk	nk	PROPN
ejpam-5974	121	5	+	+	PROPN
ejpam-5974	121	6	1)nk	1)nk	PROPN
ejpam-5974	121	7	=	=	SYM
ejpam-5974	121	8	2	2	NUM
ejpam-5974	121	9	nk	nk	NOUN
ejpam-5974	121	10	+	+	NOUN
ejpam-5974	121	11	1	1	NUM
ejpam-5974	121	12	.	.	PUNCT
ejpam-5974	121	13	illustration	illustration	NOUN
ejpam-5974	121	14	:	:	PUNCT
ejpam-5974	121	15	consider	consider	VERB
ejpam-5974	121	16	figure	figure	NOUN
ejpam-5974	121	17	5	5	NUM
ejpam-5974	121	18	,	,	PUNCT
ejpam-5974	121	19	it	it	PRON
ejpam-5974	121	20	shows	show	VERB
ejpam-5974	121	21	that	that	SCONJ
ejpam-5974	121	22	the	the	DET
ejpam-5974	121	23	number	number	NOUN
ejpam-5974	121	24	of	of	ADP
ejpam-5974	121	25	edges	edge	NOUN
ejpam-5974	121	26	and	and	CCONJ
ejpam-5974	121	27	vertices	vertex	NOUN
ejpam-5974	121	28	of	of	ADP
ejpam-5974	121	29	b3,4	b3,4	ADJ
ejpam-5974	121	30	are	be	AUX
ejpam-5974	121	31	|e(b3,4)|	|e(b3,4)|	PROPN
ejpam-5974	121	32	=	=	SYM
ejpam-5974	121	33	12	12	NUM
ejpam-5974	121	34	and	and	CCONJ
ejpam-5974	121	35	|v	|v	PROPN
ejpam-5974	121	36	(	(	PUNCT
ejpam-5974	121	37	b3,4)|	b3,4)|	VERB
ejpam-5974	121	38	=	=	SYM
ejpam-5974	121	39	13	13	NUM
ejpam-5974	121	40	respectively	respectively	ADV
ejpam-5974	121	41	.	.	PUNCT
ejpam-5974	122	1	clearly	clearly	ADV
ejpam-5974	122	2	,	,	PUNCT
ejpam-5974	122	3	d(b3,4	d(b3,4	ADJ
ejpam-5974	122	4	)	)	PUNCT
ejpam-5974	122	5	=	=	SYM
ejpam-5974	122	6	2|e(b3,4)|	2|e(b3,4)|	PROPN
ejpam-5974	122	7	|v	|v	X
ejpam-5974	122	8	(	(	PUNCT
ejpam-5974	122	9	b3,4)|(|v	b3,4)|(|v	X
ejpam-5974	122	10	(	(	PUNCT
ejpam-5974	122	11	b3,4)|	b3,4)|	VERB
ejpam-5974	122	12	−	−	NOUN
ejpam-5974	122	13	1	1	NUM
ejpam-5974	122	14	)	)	PUNCT
ejpam-5974	122	15	=	=	PUNCT
ejpam-5974	123	1	2|12|	2|12|	NUM
ejpam-5974	123	2	13(13−	13(13−	NUM
ejpam-5974	123	3	1	1	NUM
ejpam-5974	123	4	)	)	PUNCT
ejpam-5974	123	5	=	=	SYM
ejpam-5974	123	6	24	24	NUM
ejpam-5974	123	7	13(12	13(12	NUM
ejpam-5974	123	8	)	)	PUNCT
ejpam-5974	123	9	=	=	SYM
ejpam-5974	124	1	24	24	NUM
ejpam-5974	124	2	156	156	NUM
ejpam-5974	124	3	=	=	SYM
ejpam-5974	124	4	0.153	0.153	NUM
ejpam-5974	124	5	or	or	CCONJ
ejpam-5974	124	6	d(b3,4	d(b3,4	ADJ
ejpam-5974	124	7	)	)	PUNCT
ejpam-5974	124	8	=	=	SYM
ejpam-5974	124	9	2	2	NUM
ejpam-5974	124	10	nk	nk	NOUN
ejpam-5974	124	11	+	+	NOUN
ejpam-5974	124	12	1	1	NUM
ejpam-5974	124	13	=	=	SYM
ejpam-5974	124	14	2	2	NUM
ejpam-5974	124	15	3(4	3(4	NUM
ejpam-5974	124	16	)	)	PUNCT
ejpam-5974	125	1	+	+	CCONJ
ejpam-5974	125	2	1	1	NUM
ejpam-5974	125	3	=	=	SYM
ejpam-5974	125	4	2	2	NUM
ejpam-5974	125	5	13	13	NUM
ejpam-5974	125	6	=	=	SYM
ejpam-5974	125	7	0.153	0.153	NUM
ejpam-5974	125	8	.	.	PUNCT
ejpam-5974	125	9	r.	r.	PROPN
ejpam-5974	125	10	sango	sango	PROPN
ejpam-5974	125	11	,	,	PUNCT
ejpam-5974	125	12	i.	i.	NOUN
ejpam-5974	125	13	cabahug	cabahug	PROPN
ejpam-5974	125	14	,	,	PUNCT
ejpam-5974	125	15	jr	jr	PROPN
ejpam-5974	125	16	/	/	SYM
ejpam-5974	125	17	eur	eur	PROPN
ejpam-5974	125	18	.	.	PUNCT
ejpam-5974	126	1	j.	j.	PROPN
ejpam-5974	126	2	pure	pure	PROPN
ejpam-5974	126	3	appl	appl	PROPN
ejpam-5974	126	4	.	.	PROPN
ejpam-5974	126	5	math	math	PROPN
ejpam-5974	126	6	,	,	PUNCT
ejpam-5974	126	7	18	18	NUM
ejpam-5974	126	8	(	(	PUNCT
ejpam-5974	126	9	3	3	NUM
ejpam-5974	126	10	)	)	PUNCT
ejpam-5974	126	11	(	(	PUNCT
ejpam-5974	126	12	2025	2025	NUM
ejpam-5974	126	13	)	)	PUNCT
ejpam-5974	126	14	,	,	PUNCT
ejpam-5974	126	15	5974	5974	NUM
ejpam-5974	126	16	7	7	NUM
ejpam-5974	126	17	of	of	ADP
ejpam-5974	126	18	16	16	NUM
ejpam-5974	126	19	v0	v0	NOUN
ejpam-5974	126	20	v1	v1	NOUN
ejpam-5974	126	21	v2	v2	PROPN
ejpam-5974	126	22	v3	v3	PROPN
ejpam-5974	126	23	v6	v6	PROPN
ejpam-5974	126	24	v4	v4	PROPN
ejpam-5974	126	25	v5	v5	PROPN
ejpam-5974	126	26	v7	v7	VERB
ejpam-5974	126	27	v9	v9	PROPN
ejpam-5974	126	28	v8	v8	PROPN
ejpam-5974	126	29	v10	v10	PROPN
ejpam-5974	126	30	v11	v11	NOUN
ejpam-5974	126	31	v	v	PROPN
ejpam-5974	126	32	figure	figure	NOUN
ejpam-5974	126	33	5	5	NUM
ejpam-5974	126	34	:	:	PUNCT
ejpam-5974	126	35	the	the	DET
ejpam-5974	126	36	banana	banana	NOUN
ejpam-5974	126	37	graph	graph	NOUN
ejpam-5974	126	38	b3,4	b3,4	ADJ
ejpam-5974	126	39	theorem	theorem	NOUN
ejpam-5974	126	40	5	5	NUM
ejpam-5974	126	41	.	.	PUNCT
ejpam-5974	127	1	let	let	VERB
ejpam-5974	127	2	g	g	PRON
ejpam-5974	127	3	be	be	AUX
ejpam-5974	127	4	a	a	DET
ejpam-5974	127	5	lollipop	lollipop	NOUN
ejpam-5974	127	6	graph	graph	NOUN
ejpam-5974	127	7	(	(	PUNCT
ejpam-5974	127	8	lm	lm	INTJ
ejpam-5974	127	9	,	,	PUNCT
ejpam-5974	127	10	n	n	CCONJ
ejpam-5974	127	11	)	)	PUNCT
ejpam-5974	127	12	where	where	SCONJ
ejpam-5974	127	13	m	m	PROPN
ejpam-5974	127	14	≥	≥	VERB
ejpam-5974	127	15	3	3	NUM
ejpam-5974	127	16	and	and	CCONJ
ejpam-5974	127	17	n	n	PRON
ejpam-5974	127	18	≥	≥	NOUN
ejpam-5974	127	19	1	1	NUM
ejpam-5974	127	20	.	.	PUNCT
ejpam-5974	128	1	then	then	ADV
ejpam-5974	128	2	d(lm	d(lm	NUM
ejpam-5974	128	3	,	,	PUNCT
ejpam-5974	128	4	n	n	CCONJ
ejpam-5974	128	5	)	)	PUNCT
ejpam-5974	128	6	=	=	SYM
ejpam-5974	128	7	m(m−	m(m−	PROPN
ejpam-5974	128	8	1	1	NUM
ejpam-5974	128	9	)	)	PUNCT
ejpam-5974	129	1	+	+	NUM
ejpam-5974	129	2	2n	2n	NUM
ejpam-5974	129	3	(	(	PUNCT
ejpam-5974	129	4	m+	m+	NOUN
ejpam-5974	129	5	n)(m+	n)(m+	X
ejpam-5974	129	6	n−	n−	NOUN
ejpam-5974	129	7	1	1	NUM
ejpam-5974	129	8	)	)	PUNCT
ejpam-5974	129	9	.	.	PUNCT
ejpam-5974	130	1	proof	proof	NOUN
ejpam-5974	130	2	.	.	PUNCT
ejpam-5974	131	1	a	a	DET
ejpam-5974	131	2	lollipop	lollipop	NOUN
ejpam-5974	131	3	graph	graph	NOUN
ejpam-5974	131	4	lm	lm	PROPN
ejpam-5974	131	5	,	,	PUNCT
ejpam-5974	131	6	n	n	PRON
ejpam-5974	131	7	is	be	AUX
ejpam-5974	131	8	constructed	construct	VERB
ejpam-5974	131	9	by	by	ADP
ejpam-5974	131	10	a	a	DET
ejpam-5974	131	11	complete	complete	ADJ
ejpam-5974	131	12	graph	graph	NOUN
ejpam-5974	131	13	kn	kn	PROPN
ejpam-5974	131	14	and	and	CCONJ
ejpam-5974	131	15	a	a	DET
ejpam-5974	131	16	path	path	NOUN
ejpam-5974	131	17	graph	graph	NOUN
ejpam-5974	131	18	pn	pn	NOUN
ejpam-5974	131	19	adding	add	VERB
ejpam-5974	131	20	exactly	exactly	ADV
ejpam-5974	131	21	one	one	NUM
ejpam-5974	131	22	edge	edge	NOUN
ejpam-5974	131	23	that	that	PRON
ejpam-5974	131	24	connect	connect	VERB
ejpam-5974	131	25	one	one	NUM
ejpam-5974	131	26	vertex	vertex	NOUN
ejpam-5974	131	27	first	first	ADV
ejpam-5974	131	28	in	in	ADP
ejpam-5974	131	29	kn	kn	PROPN
ejpam-5974	131	30	to	to	ADP
ejpam-5974	131	31	one	one	NUM
ejpam-5974	131	32	vertex	vertex	NOUN
ejpam-5974	131	33	in	in	ADP
ejpam-5974	131	34	pn	pn	PROPN
ejpam-5974	131	35	.	.	PUNCT
ejpam-5974	132	1	thus	thus	ADV
ejpam-5974	132	2	,	,	PUNCT
ejpam-5974	132	3	the	the	DET
ejpam-5974	132	4	number	number	NOUN
ejpam-5974	132	5	of	of	ADP
ejpam-5974	132	6	edges	edge	NOUN
ejpam-5974	132	7	and	and	CCONJ
ejpam-5974	132	8	vertices	vertex	NOUN
ejpam-5974	132	9	of	of	ADP
ejpam-5974	132	10	the	the	DET
ejpam-5974	132	11	lollipop	lollipop	NOUN
ejpam-5974	132	12	graph	graph	NOUN
ejpam-5974	132	13	lm	lm	PROPN
ejpam-5974	132	14	,	,	PUNCT
ejpam-5974	132	15	n	n	PRON
ejpam-5974	132	16	are	be	AUX
ejpam-5974	132	17	|e(lm	|e(lm	PROPN
ejpam-5974	132	18	,	,	PUNCT
ejpam-5974	132	19	n)|	n)|	NOUN
ejpam-5974	132	20	=	=	SYM
ejpam-5974	132	21	m(m−	m(m−	PROPN
ejpam-5974	132	22	1	1	NUM
ejpam-5974	132	23	)	)	PUNCT
ejpam-5974	133	1	+	+	NUM
ejpam-5974	133	2	2n	2n	NUM
ejpam-5974	133	3	2	2	NUM
ejpam-5974	133	4	and	and	CCONJ
ejpam-5974	133	5	|v	|v	PROPN
ejpam-5974	133	6	(	(	PUNCT
ejpam-5974	133	7	lm	lm	PROPN
ejpam-5974	133	8	,	,	PUNCT
ejpam-5974	133	9	n)|	n)|	NOUN
ejpam-5974	133	10	=	=	SYM
ejpam-5974	133	11	m+	m+	NUM
ejpam-5974	133	12	n	n	NUM
ejpam-5974	133	13	respectively	respectively	ADV
ejpam-5974	133	14	.	.	PUNCT
ejpam-5974	134	1	therefore	therefore	ADV
ejpam-5974	134	2	,	,	PUNCT
ejpam-5974	134	3	d(lm	d(lm	NOUN
ejpam-5974	134	4	,	,	PUNCT
ejpam-5974	134	5	n	n	CCONJ
ejpam-5974	134	6	)	)	PUNCT
ejpam-5974	134	7	=	=	SYM
ejpam-5974	134	8	2|e(lm	2|e(lm	PROPN
ejpam-5974	134	9	,	,	PUNCT
ejpam-5974	134	10	n)|	n)|	PROPN
ejpam-5974	134	11	|v	|v	PROPN
ejpam-5974	134	12	(	(	PUNCT
ejpam-5974	134	13	lm	lm	INTJ
ejpam-5974	134	14	,	,	PUNCT
ejpam-5974	134	15	n)|(|v	n)|(|v	X
ejpam-5974	134	16	(	(	PUNCT
ejpam-5974	134	17	lm	lm	INTJ
ejpam-5974	134	18	,	,	PUNCT
ejpam-5974	134	19	n)|	n)|	NOUN
ejpam-5974	134	20	−	−	PROPN
ejpam-5974	134	21	1	1	NUM
ejpam-5974	134	22	)	)	PUNCT
ejpam-5974	134	23	=	=	SYM
ejpam-5974	134	24	2	2	NUM
ejpam-5974	134	25	[	[	PUNCT
ejpam-5974	134	26	m(m−	m(m−	X
ejpam-5974	134	27	1	1	NUM
ejpam-5974	134	28	)	)	PUNCT
ejpam-5974	134	29	+	+	NUM
ejpam-5974	134	30	2n	2n	NUM
ejpam-5974	134	31	2	2	NUM
ejpam-5974	134	32	]	]	PUNCT
ejpam-5974	134	33	(	(	PUNCT
ejpam-5974	134	34	m+	m+	NUM
ejpam-5974	134	35	n)(m+	n)(m+	X
ejpam-5974	134	36	n+	n+	NUM
ejpam-5974	134	37	1	1	X
ejpam-5974	134	38	)	)	PUNCT
ejpam-5974	134	39	=	=	PUNCT
ejpam-5974	134	40	m(m−	m(m−	PROPN
ejpam-5974	134	41	1	1	NUM
ejpam-5974	134	42	)	)	PUNCT
ejpam-5974	134	43	+	+	NUM
ejpam-5974	134	44	2n	2n	NUM
ejpam-5974	134	45	(	(	PUNCT
ejpam-5974	134	46	m+	m+	NUM
ejpam-5974	134	47	n)(m+	n)(m+	X
ejpam-5974	134	48	n+	n+	NUM
ejpam-5974	134	49	1	1	NUM
ejpam-5974	134	50	)	)	PUNCT
ejpam-5974	134	51	.	.	PUNCT
ejpam-5974	135	1	illustration	illustration	NOUN
ejpam-5974	135	2	:	:	PUNCT
ejpam-5974	135	3	consider	consider	VERB
ejpam-5974	135	4	figure	figure	NOUN
ejpam-5974	135	5	6	6	NUM
ejpam-5974	135	6	,	,	PUNCT
ejpam-5974	135	7	it	it	PRON
ejpam-5974	135	8	shows	show	VERB
ejpam-5974	135	9	that	that	SCONJ
ejpam-5974	135	10	the	the	DET
ejpam-5974	135	11	number	number	NOUN
ejpam-5974	135	12	edges	edge	VERB
ejpam-5974	135	13	and	and	CCONJ
ejpam-5974	135	14	vertices	vertex	NOUN
ejpam-5974	135	15	of	of	ADP
ejpam-5974	135	16	l4,3	l4,3	PROPN
ejpam-5974	135	17	are	be	AUX
ejpam-5974	135	18	|e(l4,3)|	|e(l4,3)|	NOUN
ejpam-5974	135	19	=	=	SYM
ejpam-5974	135	20	9	9	NUM
ejpam-5974	135	21	and	and	CCONJ
ejpam-5974	135	22	|v	|v	PROPN
ejpam-5974	135	23	(	(	PUNCT
ejpam-5974	135	24	l4,3)|	l4,3)|	NOUN
ejpam-5974	135	25	=	=	SYM
ejpam-5974	135	26	7	7	X
ejpam-5974	135	27	.	.	PUNCT
ejpam-5974	135	28	clearly	clearly	ADV
ejpam-5974	135	29	,	,	PUNCT
ejpam-5974	135	30	d(l4,3	d(l4,3	PROPN
ejpam-5974	135	31	)	)	PUNCT
ejpam-5974	135	32	=	=	SYM
ejpam-5974	135	33	2|e(l4,3)|	2|e(l4,3)|	NUM
ejpam-5974	135	34	|v	|v	NOUN
ejpam-5974	135	35	(	(	PUNCT
ejpam-5974	135	36	l4,3)|(|v	l4,3)|(|v	X
ejpam-5974	135	37	(	(	PUNCT
ejpam-5974	135	38	l4,3)|	l4,3)|	NOUN
ejpam-5974	135	39	−	−	NOUN
ejpam-5974	135	40	1	1	NUM
ejpam-5974	135	41	)	)	PUNCT
ejpam-5974	135	42	2|9|	2|9|	NUM
ejpam-5974	135	43	7(7−	7(7−	NUM
ejpam-5974	135	44	1	1	NUM
ejpam-5974	135	45	)	)	PUNCT
ejpam-5974	135	46	=	=	SYM
ejpam-5974	135	47	18	18	NUM
ejpam-5974	135	48	7(6	7(6	NUM
ejpam-5974	135	49	)	)	PUNCT
ejpam-5974	135	50	=	=	SYM
ejpam-5974	135	51	18	18	NUM
ejpam-5974	135	52	42	42	NUM
ejpam-5974	135	53	=	=	SYM
ejpam-5974	135	54	0.43	0.43	NUM
ejpam-5974	135	55	or	or	CCONJ
ejpam-5974	135	56	d(l4,3	d(l4,3	PRON
ejpam-5974	135	57	)	)	PUNCT
ejpam-5974	135	58	=	=	PUNCT
ejpam-5974	135	59	m(m−	m(m−	PROPN
ejpam-5974	135	60	1	1	NUM
ejpam-5974	135	61	)	)	PUNCT
ejpam-5974	135	62	+	+	NUM
ejpam-5974	135	63	2n	2n	NUM
ejpam-5974	135	64	(	(	PUNCT
ejpam-5974	135	65	m+	m+	NUM
ejpam-5974	135	66	n)(m+	n)(m+	X
ejpam-5974	135	67	n+	n+	NUM
ejpam-5974	135	68	1	1	X
ejpam-5974	135	69	)	)	PUNCT
ejpam-5974	135	70	=	=	SYM
ejpam-5974	135	71	4(4−	4(4−	NUM
ejpam-5974	135	72	1	1	NUM
ejpam-5974	135	73	)	)	PUNCT
ejpam-5974	135	74	+	+	CCONJ
ejpam-5974	135	75	2(3	2(3	NUM
ejpam-5974	135	76	)	)	PUNCT
ejpam-5974	135	77	(	(	PUNCT
ejpam-5974	135	78	4	4	NUM
ejpam-5974	135	79	+	+	SYM
ejpam-5974	135	80	3)(4	3)(4	NUM
ejpam-5974	135	81	+	+	CCONJ
ejpam-5974	135	82	3−	3−	NUM
ejpam-5974	135	83	1	1	NUM
ejpam-5974	135	84	)	)	PUNCT
ejpam-5974	135	85	=	=	SYM
ejpam-5974	135	86	18	18	NUM
ejpam-5974	135	87	42	42	NUM
ejpam-5974	135	88	=	=	SYM
ejpam-5974	135	89	0.43	0.43	NUM
ejpam-5974	135	90	.	.	PUNCT
ejpam-5974	136	1	v1	v1	PROPN
ejpam-5974	136	2	v2	v2	PROPN
ejpam-5974	136	3	v3	v3	PROPN
ejpam-5974	136	4	v4	v4	PROPN
ejpam-5974	136	5	v5	v5	PROPN
ejpam-5974	136	6	v6	v6	NOUN
ejpam-5974	136	7	v7	v7	NOUN
ejpam-5974	136	8	figure	figure	NOUN
ejpam-5974	136	9	6	6	NUM
ejpam-5974	136	10	:	:	PUNCT
ejpam-5974	136	11	the	the	DET
ejpam-5974	136	12	lollipop	lollipop	NOUN
ejpam-5974	136	13	graph	graph	VERB
ejpam-5974	136	14	l4,3	l4,3	PROPN
ejpam-5974	136	15	r.	r.	NOUN
ejpam-5974	136	16	sango	sango	PROPN
ejpam-5974	136	17	,	,	PUNCT
ejpam-5974	136	18	i.	i.	NOUN
ejpam-5974	136	19	cabahug	cabahug	PROPN
ejpam-5974	136	20	,	,	PUNCT
ejpam-5974	136	21	jr	jr	PROPN
ejpam-5974	136	22	/	/	SYM
ejpam-5974	136	23	eur	eur	PROPN
ejpam-5974	136	24	.	.	PUNCT
ejpam-5974	137	1	j.	j.	PROPN
ejpam-5974	137	2	pure	pure	PROPN
ejpam-5974	137	3	appl	appl	PROPN
ejpam-5974	137	4	.	.	PROPN
ejpam-5974	137	5	math	math	PROPN
ejpam-5974	137	6	,	,	PUNCT
ejpam-5974	137	7	18	18	NUM
ejpam-5974	137	8	(	(	PUNCT
ejpam-5974	137	9	3	3	NUM
ejpam-5974	137	10	)	)	PUNCT
ejpam-5974	137	11	(	(	PUNCT
ejpam-5974	137	12	2025	2025	NUM
ejpam-5974	137	13	)	)	PUNCT
ejpam-5974	137	14	,	,	PUNCT
ejpam-5974	137	15	5974	5974	NUM
ejpam-5974	137	16	8	8	NUM
ejpam-5974	137	17	of	of	ADP
ejpam-5974	137	18	16	16	NUM
ejpam-5974	137	19	theorem	theorem	NOUN
ejpam-5974	137	20	6	6	NUM
ejpam-5974	137	21	.	.	PUNCT
ejpam-5974	138	1	let	let	VERB
ejpam-5974	138	2	g	g	PRON
ejpam-5974	138	3	be	be	AUX
ejpam-5974	138	4	a	a	DET
ejpam-5974	138	5	tadpole	tadpole	NOUN
ejpam-5974	138	6	graph	graph	NOUN
ejpam-5974	138	7	(	(	PUNCT
ejpam-5974	138	8	tm	tm	NOUN
ejpam-5974	138	9	,	,	PUNCT
ejpam-5974	138	10	n	n	CCONJ
ejpam-5974	138	11	)	)	PUNCT
ejpam-5974	138	12	where	where	SCONJ
ejpam-5974	138	13	m	m	PROPN
ejpam-5974	138	14	≥	≥	VERB
ejpam-5974	138	15	3	3	NUM
ejpam-5974	138	16	and	and	CCONJ
ejpam-5974	138	17	n	n	PRON
ejpam-5974	138	18	≥	≥	NOUN
ejpam-5974	138	19	1	1	NUM
ejpam-5974	138	20	.	.	PUNCT
ejpam-5974	139	1	then	then	ADV
ejpam-5974	139	2	d(tm	d(tm	PROPN
ejpam-5974	139	3	,	,	PUNCT
ejpam-5974	139	4	n	n	CCONJ
ejpam-5974	139	5	)	)	PUNCT
ejpam-5974	140	1	=	=	SYM
ejpam-5974	140	2	2	2	NUM
ejpam-5974	140	3	m+	m+	NUM
ejpam-5974	140	4	n−	n−	NOUN
ejpam-5974	140	5	1	1	NUM
ejpam-5974	140	6	.	.	PUNCT
ejpam-5974	141	1	proof	proof	NOUN
ejpam-5974	141	2	.	.	PUNCT
ejpam-5974	142	1	a	a	DET
ejpam-5974	142	2	tadpole	tadpole	PROPN
ejpam-5974	142	3	graph	graph	NOUN
ejpam-5974	142	4	tm	tm	PROPN
ejpam-5974	142	5	,	,	PUNCT
ejpam-5974	142	6	n	n	PRON
ejpam-5974	142	7	is	be	AUX
ejpam-5974	142	8	constructed	construct	VERB
ejpam-5974	142	9	by	by	ADP
ejpam-5974	142	10	cycle	cycle	NOUN
ejpam-5974	142	11	graph	graph	NOUN
ejpam-5974	142	12	cn	cn	PROPN
ejpam-5974	142	13	and	and	CCONJ
ejpam-5974	142	14	path	path	NOUN
ejpam-5974	142	15	graph	graph	NOUN
ejpam-5974	142	16	pn	pn	NOUN
ejpam-5974	142	17	adding	add	VERB
ejpam-5974	142	18	exactly	exactly	ADV
ejpam-5974	142	19	one	one	NUM
ejpam-5974	142	20	edge	edge	NOUN
ejpam-5974	142	21	that	that	PRON
ejpam-5974	142	22	connect	connect	VERB
ejpam-5974	142	23	one	one	NUM
ejpam-5974	142	24	vertex	vertex	NOUN
ejpam-5974	142	25	in	in	ADP
ejpam-5974	142	26	cn	cn	PROPN
ejpam-5974	142	27	to	to	ADP
ejpam-5974	142	28	one	one	NUM
ejpam-5974	142	29	vertex	vertex	NOUN
ejpam-5974	142	30	in	in	ADP
ejpam-5974	142	31	pn	pn	PROPN
ejpam-5974	142	32	.	.	PUNCT
ejpam-5974	143	1	thus	thus	ADV
ejpam-5974	143	2	,	,	PUNCT
ejpam-5974	143	3	the	the	DET
ejpam-5974	143	4	number	number	NOUN
ejpam-5974	143	5	of	of	ADP
ejpam-5974	143	6	vertices	vertex	NOUN
ejpam-5974	143	7	and	and	CCONJ
ejpam-5974	143	8	edges	edge	NOUN
ejpam-5974	143	9	of	of	ADP
ejpam-5974	143	10	the	the	DET
ejpam-5974	143	11	tadpole	tadpole	NOUN
ejpam-5974	143	12	graph	graph	NOUN
ejpam-5974	143	13	tm	tm	PROPN
ejpam-5974	143	14	,	,	PUNCT
ejpam-5974	143	15	n	n	X
ejpam-5974	143	16	is	be	AUX
ejpam-5974	143	17	|e(tm	|e(tm	NOUN
ejpam-5974	143	18	,	,	PUNCT
ejpam-5974	143	19	n)|	n)|	NOUN
ejpam-5974	143	20	=	=	SYM
ejpam-5974	143	21	|v	|v	PROPN
ejpam-5974	143	22	(	(	PUNCT
ejpam-5974	143	23	tm	tm	PROPN
ejpam-5974	143	24	,	,	PUNCT
ejpam-5974	143	25	n)|	n)|	NOUN
ejpam-5974	143	26	=	=	SYM
ejpam-5974	143	27	m+n	m+n	PROPN
ejpam-5974	143	28	.	.	PUNCT
ejpam-5974	144	1	therefore	therefore	ADV
ejpam-5974	144	2	,	,	PUNCT
ejpam-5974	144	3	d(tm	d(tm	PROPN
ejpam-5974	144	4	,	,	PUNCT
ejpam-5974	144	5	n	n	CCONJ
ejpam-5974	144	6	)	)	PUNCT
ejpam-5974	144	7	=	=	SYM
ejpam-5974	145	1	2|e(tm	2|e(tm	NOUN
ejpam-5974	145	2	,	,	PUNCT
ejpam-5974	145	3	n)|	n)|	PROPN
ejpam-5974	145	4	|v	|v	PROPN
ejpam-5974	145	5	(	(	PUNCT
ejpam-5974	145	6	tm	tm	PROPN
ejpam-5974	145	7	,	,	PUNCT
ejpam-5974	145	8	n)|(|v	n)|(|v	PROPN
ejpam-5974	145	9	(	(	PUNCT
ejpam-5974	145	10	tm	tm	NOUN
ejpam-5974	145	11	,	,	PUNCT
ejpam-5974	145	12	n)|	n)|	NOUN
ejpam-5974	145	13	−	−	PROPN
ejpam-5974	145	14	1	1	NUM
ejpam-5974	145	15	)	)	PUNCT
ejpam-5974	145	16	=	=	SYM
ejpam-5974	145	17	2(m+	2(m+	NUM
ejpam-5974	145	18	n	n	CCONJ
ejpam-5974	145	19	)	)	PUNCT
ejpam-5974	145	20	(	(	PUNCT
ejpam-5974	145	21	m+	m+	NOUN
ejpam-5974	145	22	n)(m+	n)(m+	X
ejpam-5974	145	23	n−	n−	NOUN
ejpam-5974	145	24	1	1	NUM
ejpam-5974	145	25	)	)	PUNCT
ejpam-5974	145	26	=	=	SYM
ejpam-5974	145	27	2	2	NUM
ejpam-5974	145	28	m+	m+	NUM
ejpam-5974	145	29	n−	n−	NOUN
ejpam-5974	145	30	1	1	NUM
ejpam-5974	145	31	.	.	PUNCT
ejpam-5974	145	32	consider	consider	VERB
ejpam-5974	145	33	figure	figure	NOUN
ejpam-5974	145	34	7	7	NUM
ejpam-5974	145	35	,	,	PUNCT
ejpam-5974	145	36	it	it	PRON
ejpam-5974	145	37	shows	show	VERB
ejpam-5974	145	38	that	that	SCONJ
ejpam-5974	145	39	the	the	DET
ejpam-5974	145	40	number	number	NOUN
ejpam-5974	145	41	edges	edge	VERB
ejpam-5974	145	42	and	and	CCONJ
ejpam-5974	145	43	vertices	vertex	NOUN
ejpam-5974	145	44	of	of	ADP
ejpam-5974	145	45	t4,3	t4,3	PROPN
ejpam-5974	145	46	are	be	AUX
ejpam-5974	145	47	|e(t4,3)|	|e(t4,3)|	NOUN
ejpam-5974	145	48	=	=	SYM
ejpam-5974	145	49	7	7	NUM
ejpam-5974	145	50	and	and	CCONJ
ejpam-5974	145	51	|v	|v	PROPN
ejpam-5974	145	52	(	(	PUNCT
ejpam-5974	145	53	t4,3)|	t4,3)|	NOUN
ejpam-5974	145	54	=	=	NOUN
ejpam-5974	145	55	7	7	NUM
ejpam-5974	145	56	respectively	respectively	ADV
ejpam-5974	145	57	.	.	PUNCT
ejpam-5974	146	1	clearly	clearly	ADV
ejpam-5974	146	2	,	,	PUNCT
ejpam-5974	146	3	d(t4,3	d(t4,3	PROPN
ejpam-5974	146	4	)	)	PUNCT
ejpam-5974	146	5	=	=	SYM
ejpam-5974	147	1	2|e(t4,3)|	2|e(t4,3)|	NUM
ejpam-5974	147	2	|v	|v	X
ejpam-5974	147	3	(	(	PUNCT
ejpam-5974	147	4	t4,3)|(|v	t4,3)|(|v	X
ejpam-5974	147	5	(	(	PUNCT
ejpam-5974	147	6	t4,3)|	t4,3)|	NOUN
ejpam-5974	147	7	−	−	NOUN
ejpam-5974	147	8	1	1	NUM
ejpam-5974	147	9	)	)	PUNCT
ejpam-5974	147	10	=	=	PUNCT
ejpam-5974	148	1	2|7|	2|7|	X
ejpam-5974	148	2	7(7−	7(7−	NUM
ejpam-5974	148	3	1	1	NUM
ejpam-5974	148	4	)	)	PUNCT
ejpam-5974	148	5	=	=	SYM
ejpam-5974	148	6	14	14	NUM
ejpam-5974	148	7	7(6	7(6	NUM
ejpam-5974	148	8	)	)	PUNCT
ejpam-5974	148	9	=	=	SYM
ejpam-5974	149	1	14	14	NUM
ejpam-5974	149	2	42	42	NUM
ejpam-5974	149	3	=	=	SYM
ejpam-5974	149	4	0.33	0.33	NUM
ejpam-5974	149	5	or	or	CCONJ
ejpam-5974	149	6	d(l4,3	d(l4,3	PRON
ejpam-5974	149	7	)	)	PUNCT
ejpam-5974	149	8	=	=	SYM
ejpam-5974	149	9	2	2	NUM
ejpam-5974	149	10	m+	m+	NUM
ejpam-5974	149	11	n−	n−	NOUN
ejpam-5974	149	12	1	1	NUM
ejpam-5974	149	13	=	=	SYM
ejpam-5974	149	14	2	2	NUM
ejpam-5974	149	15	(	(	PUNCT
ejpam-5974	149	16	4	4	NUM
ejpam-5974	149	17	+	+	CCONJ
ejpam-5974	149	18	3−	3−	NUM
ejpam-5974	149	19	1	1	NUM
ejpam-5974	149	20	)	)	PUNCT
ejpam-5974	149	21	)	)	PUNCT
ejpam-5974	150	1	=	=	SYM
ejpam-5974	150	2	2	2	NUM
ejpam-5974	150	3	6	6	NUM
ejpam-5974	150	4	=	=	SYM
ejpam-5974	150	5	0.33	0.33	NUM
ejpam-5974	150	6	.	.	PUNCT
ejpam-5974	151	1	figure	figure	NOUN
ejpam-5974	151	2	7	7	NUM
ejpam-5974	151	3	:	:	PUNCT
ejpam-5974	151	4	the	the	DET
ejpam-5974	151	5	tadpole	tadpole	NOUN
ejpam-5974	151	6	graph	graph	NOUN
ejpam-5974	151	7	t4,3	t4,3	PROPN
ejpam-5974	151	8	theorem	theorem	NOUN
ejpam-5974	151	9	7	7	NUM
ejpam-5974	151	10	.	.	PUNCT
ejpam-5974	152	1	let	let	VERB
ejpam-5974	152	2	g	g	PRON
ejpam-5974	152	3	be	be	AUX
ejpam-5974	152	4	a	a	DET
ejpam-5974	152	5	gear	gear	NOUN
ejpam-5974	152	6	graph	graph	NOUN
ejpam-5974	152	7	gn	gn	PROPN
ejpam-5974	152	8	where	where	SCONJ
ejpam-5974	152	9	n	n	PRON
ejpam-5974	152	10	≥	≥	NOUN
ejpam-5974	152	11	3	3	NUM
ejpam-5974	152	12	.	.	PUNCT
ejpam-5974	153	1	then	then	ADV
ejpam-5974	153	2	d(gn	d(gn	NOUN
ejpam-5974	153	3	)	)	PUNCT
ejpam-5974	153	4	=	=	SYM
ejpam-5974	153	5	3	3	NUM
ejpam-5974	153	6	2n+	2n+	NUM
ejpam-5974	153	7	1	1	NUM
ejpam-5974	153	8	.	.	PUNCT
ejpam-5974	154	1	proof	proof	NOUN
ejpam-5974	154	2	.	.	PUNCT
ejpam-5974	155	1	a	a	DET
ejpam-5974	155	2	gear	gear	NOUN
ejpam-5974	155	3	graph	graph	NOUN
ejpam-5974	155	4	is	be	AUX
ejpam-5974	155	5	constructed	construct	VERB
ejpam-5974	155	6	by	by	ADP
ejpam-5974	155	7	wheel	wheel	NOUN
ejpam-5974	155	8	graph	graph	NOUN
ejpam-5974	155	9	wn	wn	NOUN
ejpam-5974	155	10	adding	add	VERB
ejpam-5974	155	11	vertex	vertex	NOUN
ejpam-5974	155	12	in	in	ADP
ejpam-5974	155	13	every	every	DET
ejpam-5974	155	14	pair	pair	NOUN
ejpam-5974	155	15	of	of	ADP
ejpam-5974	155	16	adjacent	adjacent	ADJ
ejpam-5974	155	17	in	in	ADP
ejpam-5974	155	18	the	the	DET
ejpam-5974	155	19	outer	outer	ADJ
ejpam-5974	155	20	cycle	cycle	NOUN
ejpam-5974	155	21	.	.	PUNCT
ejpam-5974	156	1	thus	thus	ADV
ejpam-5974	156	2	,	,	PUNCT
ejpam-5974	156	3	the	the	DET
ejpam-5974	156	4	number	number	NOUN
ejpam-5974	156	5	of	of	ADP
ejpam-5974	156	6	edges	edge	NOUN
ejpam-5974	156	7	and	and	CCONJ
ejpam-5974	156	8	vertices	vertex	NOUN
ejpam-5974	156	9	of	of	ADP
ejpam-5974	156	10	the	the	DET
ejpam-5974	156	11	gear	gear	NOUN
ejpam-5974	156	12	graph	graph	NOUN
ejpam-5974	156	13	gn	gn	PROPN
ejpam-5974	156	14	are	be	AUX
ejpam-5974	156	15	|e(gn)|	|e(gn)|	NUM
ejpam-5974	156	16	=	=	SYM
ejpam-5974	156	17	3n	3n	NOUN
ejpam-5974	156	18	and	and	CCONJ
ejpam-5974	156	19	the	the	DET
ejpam-5974	156	20	order	order	NOUN
ejpam-5974	156	21	is	be	AUX
ejpam-5974	156	22	|v	|v	PROPN
ejpam-5974	156	23	(	(	PUNCT
ejpam-5974	156	24	gn)|	gn)|	PROPN
ejpam-5974	156	25	=	=	SYM
ejpam-5974	156	26	2n+	2n+	NUM
ejpam-5974	156	27	1	1	NUM
ejpam-5974	156	28	respectively	respectively	ADV
ejpam-5974	156	29	.	.	PUNCT
ejpam-5974	157	1	therefore	therefore	ADV
ejpam-5974	157	2	,	,	PUNCT
ejpam-5974	157	3	d(gn	d(gn	NOUN
ejpam-5974	157	4	)	)	PUNCT
ejpam-5974	157	5	=	=	SYM
ejpam-5974	157	6	2|e(gn)|	2|e(gn)|	NUM
ejpam-5974	157	7	|v	|v	PROPN
ejpam-5974	157	8	(	(	PUNCT
ejpam-5974	157	9	gn)|(|v	gn)|(|v	PROPN
ejpam-5974	157	10	(	(	PUNCT
ejpam-5974	157	11	gn)|	gn)|	PROPN
ejpam-5974	157	12	−	−	PROPN
ejpam-5974	157	13	1	1	NUM
ejpam-5974	157	14	)	)	PUNCT
ejpam-5974	157	15	=	=	SYM
ejpam-5974	157	16	2(3n	2(3n	NUM
ejpam-5974	157	17	)	)	PUNCT
ejpam-5974	157	18	(	(	PUNCT
ejpam-5974	157	19	2n+	2n+	NUM
ejpam-5974	157	20	1)(2n+	1)(2n+	NUM
ejpam-5974	157	21	1−	1−	NUM
ejpam-5974	157	22	1	1	NUM
ejpam-5974	157	23	)	)	PUNCT
ejpam-5974	157	24	=	=	SYM
ejpam-5974	157	25	2(3n	2(3n	NUM
ejpam-5974	157	26	)	)	PUNCT
ejpam-5974	157	27	(	(	PUNCT
ejpam-5974	157	28	2n+	2n+	NUM
ejpam-5974	157	29	1)2n	1)2n	NUM
ejpam-5974	157	30	=	=	SYM
ejpam-5974	157	31	3	3	NUM
ejpam-5974	157	32	2n+	2n+	NUM
ejpam-5974	157	33	1	1	NUM
ejpam-5974	157	34	.	.	PUNCT
ejpam-5974	158	1	r.	r.	PROPN
ejpam-5974	158	2	sango	sango	PROPN
ejpam-5974	158	3	,	,	PUNCT
ejpam-5974	158	4	i.	i.	NOUN
ejpam-5974	158	5	cabahug	cabahug	PROPN
ejpam-5974	158	6	,	,	PUNCT
ejpam-5974	158	7	jr	jr	PROPN
ejpam-5974	158	8	/	/	SYM
ejpam-5974	158	9	eur	eur	PROPN
ejpam-5974	158	10	.	.	PUNCT
ejpam-5974	159	1	j.	j.	PROPN
ejpam-5974	159	2	pure	pure	PROPN
ejpam-5974	159	3	appl	appl	PROPN
ejpam-5974	159	4	.	.	PROPN
ejpam-5974	159	5	math	math	PROPN
ejpam-5974	159	6	,	,	PUNCT
ejpam-5974	159	7	18	18	NUM
ejpam-5974	159	8	(	(	PUNCT
ejpam-5974	159	9	3	3	NUM
ejpam-5974	159	10	)	)	PUNCT
ejpam-5974	159	11	(	(	PUNCT
ejpam-5974	159	12	2025	2025	NUM
ejpam-5974	159	13	)	)	PUNCT
ejpam-5974	159	14	,	,	PUNCT
ejpam-5974	159	15	5974	5974	NUM
ejpam-5974	159	16	9	9	NUM
ejpam-5974	159	17	of	of	ADP
ejpam-5974	159	18	16	16	NUM
ejpam-5974	159	19	illustration	illustration	NOUN
ejpam-5974	159	20	:	:	PUNCT
ejpam-5974	159	21	consider	consider	VERB
ejpam-5974	159	22	figure	figure	NOUN
ejpam-5974	159	23	8	8	NUM
ejpam-5974	159	24	,	,	PUNCT
ejpam-5974	159	25	it	it	PRON
ejpam-5974	159	26	shows	show	VERB
ejpam-5974	159	27	that	that	SCONJ
ejpam-5974	159	28	the	the	DET
ejpam-5974	159	29	number	number	NOUN
ejpam-5974	159	30	of	of	ADP
ejpam-5974	159	31	edges	edge	NOUN
ejpam-5974	159	32	and	and	CCONJ
ejpam-5974	159	33	vertices	vertex	NOUN
ejpam-5974	159	34	of	of	ADP
ejpam-5974	159	35	g4	g4	NOUN
ejpam-5974	159	36	are	be	AUX
ejpam-5974	159	37	|e(g4)|	|e(g4)|	PROPN
ejpam-5974	159	38	=	=	SYM
ejpam-5974	159	39	16	16	NUM
ejpam-5974	159	40	and	and	CCONJ
ejpam-5974	159	41	|v	|v	PROPN
ejpam-5974	159	42	(	(	PUNCT
ejpam-5974	159	43	g4)|	g4)|	NOUN
ejpam-5974	159	44	=	=	NOUN
ejpam-5974	159	45	9	9	NUM
ejpam-5974	159	46	respectively	respectively	ADV
ejpam-5974	159	47	.	.	PUNCT
ejpam-5974	160	1	clearly	clearly	ADV
ejpam-5974	160	2	,	,	PUNCT
ejpam-5974	160	3	d(g4	d(g4	NOUN
ejpam-5974	160	4	)	)	PUNCT
ejpam-5974	161	1	=	=	SYM
ejpam-5974	161	2	2|e(g4)|	2|e(g4)|	NUM
ejpam-5974	161	3	|v	|v	NOUN
ejpam-5974	161	4	(	(	PUNCT
ejpam-5974	161	5	g4)|(|v	g4)|(|v	PROPN
ejpam-5974	161	6	(	(	PUNCT
ejpam-5974	161	7	g4)|	g4)|	VERB
ejpam-5974	161	8	−	−	PROPN
ejpam-5974	161	9	1	1	NUM
ejpam-5974	161	10	)	)	PUNCT
ejpam-5974	161	11	=	=	SYM
ejpam-5974	161	12	2|16|	2|16|	NUM
ejpam-5974	161	13	9(9−	9(9−	NUM
ejpam-5974	161	14	1	1	NUM
ejpam-5974	161	15	)	)	PUNCT
ejpam-5974	161	16	=	=	PUNCT
ejpam-5974	161	17	32	32	NUM
ejpam-5974	161	18	9(8	9(8	NUM
ejpam-5974	161	19	)	)	PUNCT
ejpam-5974	161	20	=	=	SYM
ejpam-5974	161	21	32	32	NUM
ejpam-5974	161	22	72	72	NUM
ejpam-5974	161	23	=	=	SYM
ejpam-5974	161	24	0.44	0.44	NUM
ejpam-5974	161	25	or	or	CCONJ
ejpam-5974	161	26	d(g4	d(g4	NOUN
ejpam-5974	161	27	)	)	PUNCT
ejpam-5974	161	28	=	=	SYM
ejpam-5974	161	29	3	3	NUM
ejpam-5974	161	30	2n+	2n+	NUM
ejpam-5974	161	31	1	1	NUM
ejpam-5974	161	32	=	=	SYM
ejpam-5974	161	33	4	4	NUM
ejpam-5974	161	34	2(4	2(4	NUM
ejpam-5974	161	35	)	)	PUNCT
ejpam-5974	161	36	+	+	CCONJ
ejpam-5974	161	37	1	1	NUM
ejpam-5974	161	38	=	=	SYM
ejpam-5974	161	39	4	4	NUM
ejpam-5974	161	40	9	9	NUM
ejpam-5974	161	41	=	=	SYM
ejpam-5974	161	42	0.44	0.44	NUM
ejpam-5974	161	43	.	.	PUNCT
ejpam-5974	162	1	v1	v1	PROPN
ejpam-5974	162	2	v2	v2	PROPN
ejpam-5974	162	3	v3	v3	PROPN
ejpam-5974	162	4	v4	v4	PROPN
ejpam-5974	162	5	c	c	PROPN
ejpam-5974	162	6	b	b	PROPN
ejpam-5974	162	7	a	a	DET
ejpam-5974	162	8	d	d	PROPN
ejpam-5974	162	9	v0	v0	NOUN
ejpam-5974	162	10	figure	figure	NOUN
ejpam-5974	162	11	8	8	NUM
ejpam-5974	162	12	:	:	PUNCT
ejpam-5974	162	13	the	the	DET
ejpam-5974	162	14	gear	gear	NOUN
ejpam-5974	162	15	graph	graph	NOUN
ejpam-5974	162	16	g4	g4	NOUN
ejpam-5974	162	17	theorem	theorem	NOUN
ejpam-5974	162	18	8	8	NUM
ejpam-5974	162	19	.	.	PUNCT
ejpam-5974	163	1	let	let	VERB
ejpam-5974	163	2	g	g	PRON
ejpam-5974	163	3	be	be	AUX
ejpam-5974	163	4	a	a	DET
ejpam-5974	163	5	wheel	wheel	NOUN
ejpam-5974	163	6	graph	graph	NOUN
ejpam-5974	163	7	(	(	PUNCT
ejpam-5974	163	8	wn	wn	PROPN
ejpam-5974	163	9	)	)	PUNCT
ejpam-5974	163	10	where	where	SCONJ
ejpam-5974	163	11	n	n	PRON
ejpam-5974	163	12	≥	≥	NOUN
ejpam-5974	163	13	4	4	NUM
ejpam-5974	163	14	.	.	PUNCT
ejpam-5974	163	15	then	then	ADV
ejpam-5974	163	16	d(wn	d(wn	PROPN
ejpam-5974	163	17	)	)	PUNCT
ejpam-5974	164	1	=	=	SYM
ejpam-5974	164	2	4	4	NUM
ejpam-5974	164	3	n	n	NOUN
ejpam-5974	164	4	.	.	PUNCT
ejpam-5974	165	1	proof	proof	NOUN
ejpam-5974	165	2	.	.	PUNCT
ejpam-5974	166	1	a	a	DET
ejpam-5974	166	2	wheel	wheel	NOUN
ejpam-5974	166	3	graph	graph	NOUN
ejpam-5974	166	4	wn	wn	PROPN
ejpam-5974	166	5	is	be	AUX
ejpam-5974	166	6	constructed	construct	VERB
ejpam-5974	166	7	by	by	ADP
ejpam-5974	166	8	cycle	cycle	NOUN
ejpam-5974	166	9	graph	graph	NOUN
ejpam-5974	166	10	cn−1	cn−1	PROPN
ejpam-5974	166	11	and	and	CCONJ
ejpam-5974	166	12	consists	consist	VERB
ejpam-5974	166	13	one	one	NUM
ejpam-5974	166	14	central	central	ADJ
ejpam-5974	166	15	vertex	vertex	NOUN
ejpam-5974	166	16	that	that	PRON
ejpam-5974	166	17	connects	connect	VERB
ejpam-5974	166	18	to	to	ADP
ejpam-5974	166	19	all	all	DET
ejpam-5974	166	20	vertices	vertex	NOUN
ejpam-5974	166	21	in	in	ADP
ejpam-5974	166	22	cn−1	cn−1	PROPN
ejpam-5974	166	23	.	.	PUNCT
ejpam-5974	167	1	thus	thus	ADV
ejpam-5974	167	2	,	,	PUNCT
ejpam-5974	167	3	the	the	DET
ejpam-5974	167	4	number	number	NOUN
ejpam-5974	167	5	of	of	ADP
ejpam-5974	167	6	edges	edge	NOUN
ejpam-5974	167	7	and	and	CCONJ
ejpam-5974	167	8	vertices	vertex	NOUN
ejpam-5974	167	9	of	of	ADP
ejpam-5974	167	10	the	the	DET
ejpam-5974	167	11	wheel	wheel	NOUN
ejpam-5974	167	12	graph	graph	NOUN
ejpam-5974	167	13	are	be	AUX
ejpam-5974	167	14	|e(wn)|	|e(wn)|	X
ejpam-5974	167	15	=	=	NOUN
ejpam-5974	167	16	2n	2n	NUM
ejpam-5974	167	17	and	and	CCONJ
ejpam-5974	167	18	order	order	NOUN
ejpam-5974	167	19	is	be	AUX
ejpam-5974	167	20	|v	|v	PROPN
ejpam-5974	167	21	(	(	PUNCT
ejpam-5974	167	22	wn)|	wn)|	X
ejpam-5974	167	23	=	=	SYM
ejpam-5974	167	24	n+	n+	PUNCT
ejpam-5974	167	25	1	1	X
ejpam-5974	167	26	.	.	PUNCT
ejpam-5974	168	1	therefore	therefore	ADV
ejpam-5974	168	2	,	,	PUNCT
ejpam-5974	168	3	d(wn	d(wn	PROPN
ejpam-5974	168	4	)	)	PUNCT
ejpam-5974	168	5	=	=	PUNCT
ejpam-5974	169	1	2|e(wn)|	2|e(wn)|	PRON
ejpam-5974	169	2	|v	|v	NOUN
ejpam-5974	169	3	(	(	PUNCT
ejpam-5974	169	4	wn)|(|v	wn)|(|v	X
ejpam-5974	169	5	(	(	PUNCT
ejpam-5974	169	6	wn)|	wn)|	NOUN
ejpam-5974	169	7	−	−	NOUN
ejpam-5974	169	8	1	1	NUM
ejpam-5974	169	9	)	)	PUNCT
ejpam-5974	169	10	=	=	SYM
ejpam-5974	169	11	2(2n	2(2n	NUM
ejpam-5974	169	12	)	)	PUNCT
ejpam-5974	169	13	(	(	PUNCT
ejpam-5974	169	14	n+	n+	NUM
ejpam-5974	169	15	1)(n+	1)(n+	NUM
ejpam-5974	169	16	1−	1−	NUM
ejpam-5974	169	17	1	1	NUM
ejpam-5974	169	18	)	)	PUNCT
ejpam-5974	169	19	=	=	NOUN
ejpam-5974	170	1	4n	4n	X
ejpam-5974	170	2	(	(	PUNCT
ejpam-5974	170	3	n+	n+	X
ejpam-5974	170	4	1)n	1)n	X
ejpam-5974	170	5	=	=	SYM
ejpam-5974	170	6	4	4	NUM
ejpam-5974	170	7	n+	n+	SYM
ejpam-5974	170	8	1	1	NUM
ejpam-5974	170	9	.	.	PUNCT
ejpam-5974	170	10	illustration	illustration	NOUN
ejpam-5974	170	11	:	:	PUNCT
ejpam-5974	170	12	consider	consider	VERB
ejpam-5974	170	13	figure	figure	NOUN
ejpam-5974	170	14	9	9	NUM
ejpam-5974	170	15	,	,	PUNCT
ejpam-5974	170	16	it	it	PRON
ejpam-5974	170	17	shows	show	VERB
ejpam-5974	170	18	that	that	SCONJ
ejpam-5974	170	19	the	the	DET
ejpam-5974	170	20	number	number	NOUN
ejpam-5974	170	21	edges	edge	VERB
ejpam-5974	170	22	and	and	CCONJ
ejpam-5974	170	23	vertices	vertex	NOUN
ejpam-5974	170	24	of	of	ADP
ejpam-5974	170	25	w5	w5	PROPN
ejpam-5974	170	26	are	be	AUX
ejpam-5974	170	27	|e(w5)|	|e(w5)|	PROPN
ejpam-5974	170	28	=	=	SYM
ejpam-5974	170	29	10	10	NUM
ejpam-5974	170	30	and	and	CCONJ
ejpam-5974	170	31	|v	|v	PROPN
ejpam-5974	170	32	(	(	PUNCT
ejpam-5974	170	33	w5)|	w5)|	X
ejpam-5974	170	34	=	=	PROPN
ejpam-5974	170	35	6	6	NUM
ejpam-5974	170	36	respectively	respectively	ADV
ejpam-5974	170	37	.	.	PUNCT
ejpam-5974	171	1	clearly	clearly	ADV
ejpam-5974	171	2	,	,	PUNCT
ejpam-5974	171	3	d(w5	d(w5	NOUN
ejpam-5974	171	4	)	)	PUNCT
ejpam-5974	172	1	=	=	SYM
ejpam-5974	172	2	2|e(w5)|	2|e(w5)|	NUM
ejpam-5974	172	3	|v	|v	NOUN
ejpam-5974	172	4	(	(	PUNCT
ejpam-5974	172	5	w5)|(|v	w5)|(|v	X
ejpam-5974	172	6	(	(	PUNCT
ejpam-5974	172	7	w5)|	w5)|	NOUN
ejpam-5974	172	8	−	−	PROPN
ejpam-5974	172	9	1	1	NUM
ejpam-5974	172	10	)	)	PUNCT
ejpam-5974	172	11	=	=	SYM
ejpam-5974	172	12	2|10|	2|10|	NUM
ejpam-5974	172	13	6(6−	6(6−	NUM
ejpam-5974	172	14	1	1	NUM
ejpam-5974	172	15	)	)	PUNCT
ejpam-5974	172	16	=	=	NOUN
ejpam-5974	172	17	20	20	NUM
ejpam-5974	172	18	6(5	6(5	NUM
ejpam-5974	172	19	)	)	PUNCT
ejpam-5974	172	20	=	=	NOUN
ejpam-5974	173	1	20	20	NUM
ejpam-5974	173	2	30	30	NUM
ejpam-5974	173	3	=	=	SYM
ejpam-5974	173	4	0.67	0.67	NUM
ejpam-5974	173	5	or	or	CCONJ
ejpam-5974	173	6	d(w5	d(w5	NOUN
ejpam-5974	173	7	)	)	PUNCT
ejpam-5974	174	1	=	=	SYM
ejpam-5974	174	2	4	4	NUM
ejpam-5974	174	3	n+	n+	SYM
ejpam-5974	174	4	1	1	NUM
ejpam-5974	174	5	=	=	SYM
ejpam-5974	174	6	4	4	NUM
ejpam-5974	174	7	5	5	NUM
ejpam-5974	174	8	+	+	CCONJ
ejpam-5974	174	9	1	1	NUM
ejpam-5974	174	10	=	=	SYM
ejpam-5974	174	11	4	4	NUM
ejpam-5974	174	12	6	6	NUM
ejpam-5974	174	13	=	=	SYM
ejpam-5974	174	14	0.67	0.67	NUM
ejpam-5974	174	15	.	.	PUNCT
ejpam-5974	174	16	r.	r.	PROPN
ejpam-5974	174	17	sango	sango	PROPN
ejpam-5974	174	18	,	,	PUNCT
ejpam-5974	174	19	i.	i.	NOUN
ejpam-5974	174	20	cabahug	cabahug	PROPN
ejpam-5974	174	21	,	,	PUNCT
ejpam-5974	174	22	jr	jr	PROPN
ejpam-5974	174	23	/	/	SYM
ejpam-5974	174	24	eur	eur	PROPN
ejpam-5974	174	25	.	.	PUNCT
ejpam-5974	175	1	j.	j.	PROPN
ejpam-5974	175	2	pure	pure	PROPN
ejpam-5974	175	3	appl	appl	PROPN
ejpam-5974	175	4	.	.	PROPN
ejpam-5974	175	5	math	math	PROPN
ejpam-5974	175	6	,	,	PUNCT
ejpam-5974	175	7	18	18	NUM
ejpam-5974	175	8	(	(	PUNCT
ejpam-5974	175	9	3	3	NUM
ejpam-5974	175	10	)	)	PUNCT
ejpam-5974	175	11	(	(	PUNCT
ejpam-5974	175	12	2025	2025	NUM
ejpam-5974	175	13	)	)	PUNCT
ejpam-5974	175	14	,	,	PUNCT
ejpam-5974	175	15	5974	5974	NUM
ejpam-5974	175	16	10	10	NUM
ejpam-5974	175	17	of	of	ADP
ejpam-5974	175	18	16	16	NUM
ejpam-5974	175	19	v1	v1	NOUN
ejpam-5974	175	20	v	v	ADP
ejpam-5974	175	21	v3	v3	PROPN
ejpam-5974	175	22	v5	v5	PROPN
ejpam-5974	175	23	v4	v4	PROPN
ejpam-5974	175	24	v2	v2	PROPN
ejpam-5974	175	25	w6	w6	PROPN
ejpam-5974	175	26	:	:	PUNCT
ejpam-5974	175	27	figure	figure	VERB
ejpam-5974	175	28	9	9	NUM
ejpam-5974	175	29	:	:	PUNCT
ejpam-5974	175	30	the	the	DET
ejpam-5974	175	31	graph	graph	NOUN
ejpam-5974	175	32	w5	w5	PROPN
ejpam-5974	175	33	theorem	theorem	NOUN
ejpam-5974	175	34	9	9	NUM
ejpam-5974	175	35	.	.	PUNCT
ejpam-5974	176	1	let	let	VERB
ejpam-5974	176	2	g	g	PRON
ejpam-5974	176	3	be	be	AUX
ejpam-5974	176	4	a	a	DET
ejpam-5974	176	5	fan	fan	NOUN
ejpam-5974	176	6	graph	graph	NOUN
ejpam-5974	176	7	graph	graph	NOUN
ejpam-5974	176	8	(	(	PUNCT
ejpam-5974	176	9	fn	fn	NOUN
ejpam-5974	176	10	)	)	PUNCT
ejpam-5974	176	11	where	where	SCONJ
ejpam-5974	176	12	n	n	PRON
ejpam-5974	176	13	≥	≥	NOUN
ejpam-5974	176	14	3	3	NUM
ejpam-5974	176	15	.	.	PUNCT
ejpam-5974	177	1	then	then	ADV
ejpam-5974	177	2	d(fn	d(fn	NOUN
ejpam-5974	177	3	)	)	PUNCT
ejpam-5974	178	1	=	=	PUNCT
ejpam-5974	179	1	4n−	4n−	NUM
ejpam-5974	179	2	2	2	NUM
ejpam-5974	179	3	(	(	PUNCT
ejpam-5974	179	4	n+	n+	NOUN
ejpam-5974	179	5	1)(n	1)(n	NUM
ejpam-5974	179	6	)	)	PUNCT
ejpam-5974	179	7	.	.	PUNCT
ejpam-5974	180	1	proof	proof	NOUN
ejpam-5974	180	2	.	.	PUNCT
ejpam-5974	181	1	a	a	DET
ejpam-5974	181	2	fan	fan	NOUN
ejpam-5974	181	3	graph	graph	NOUN
ejpam-5974	181	4	fn	fn	NOUN
ejpam-5974	181	5	is	be	AUX
ejpam-5974	181	6	constructed	construct	VERB
ejpam-5974	181	7	by	by	ADP
ejpam-5974	181	8	a	a	DET
ejpam-5974	181	9	path	path	NOUN
ejpam-5974	181	10	graph	graph	NOUN
ejpam-5974	181	11	pn	pn	NOUN
ejpam-5974	181	12	and	and	CCONJ
ejpam-5974	181	13	one	one	NUM
ejpam-5974	181	14	central	central	ADJ
ejpam-5974	181	15	vertex	vertex	NOUN
ejpam-5974	181	16	that	that	PRON
ejpam-5974	181	17	connects	connect	VERB
ejpam-5974	181	18	to	to	ADP
ejpam-5974	181	19	all	all	DET
ejpam-5974	181	20	vertices	vertex	NOUN
ejpam-5974	181	21	in	in	ADP
ejpam-5974	181	22	path	path	NOUN
ejpam-5974	181	23	graph	graph	NOUN
ejpam-5974	181	24	pn	pn	PROPN
ejpam-5974	181	25	.	.	PUNCT
ejpam-5974	182	1	thus	thus	ADV
ejpam-5974	182	2	,	,	PUNCT
ejpam-5974	182	3	the	the	DET
ejpam-5974	182	4	number	number	NOUN
ejpam-5974	182	5	of	of	ADP
ejpam-5974	182	6	edges	edge	NOUN
ejpam-5974	182	7	and	and	CCONJ
ejpam-5974	182	8	vertices	vertex	NOUN
ejpam-5974	182	9	of	of	ADP
ejpam-5974	182	10	the	the	DET
ejpam-5974	182	11	fan	fan	NOUN
ejpam-5974	182	12	graph	graph	NOUN
ejpam-5974	182	13	fn	fn	NOUN
ejpam-5974	182	14	are	be	AUX
ejpam-5974	182	15	|e(fn)|	|e(fn)|	ADJ
ejpam-5974	182	16	=	=	SYM
ejpam-5974	182	17	2n−	2n−	PROPN
ejpam-5974	182	18	1	1	NUM
ejpam-5974	182	19	and	and	CCONJ
ejpam-5974	182	20	order	order	NOUN
ejpam-5974	182	21	|v	|v	NOUN
ejpam-5974	182	22	(	(	PUNCT
ejpam-5974	182	23	fn)|	fn)|	X
ejpam-5974	182	24	=	=	PUNCT
ejpam-5974	182	25	n+	n+	ADP
ejpam-5974	182	26	1	1	NUM
ejpam-5974	182	27	respectively	respectively	ADV
ejpam-5974	182	28	.	.	PUNCT
ejpam-5974	183	1	therefore	therefore	ADV
ejpam-5974	183	2	,	,	PUNCT
ejpam-5974	183	3	d(fn	d(fn	NOUN
ejpam-5974	183	4	)	)	PUNCT
ejpam-5974	184	1	=	=	SYM
ejpam-5974	185	1	2|e(fn)|	2|e(fn)|	NUM
ejpam-5974	185	2	|v	|v	X
ejpam-5974	185	3	(	(	PUNCT
ejpam-5974	185	4	fn)|(|v	fn)|(|v	PROPN
ejpam-5974	185	5	(	(	PUNCT
ejpam-5974	185	6	fn)|	fn)|	PROPN
ejpam-5974	185	7	−	−	PROPN
ejpam-5974	185	8	1	1	NUM
ejpam-5974	185	9	)	)	PUNCT
ejpam-5974	185	10	=	=	SYM
ejpam-5974	185	11	2	2	NUM
ejpam-5974	185	12	m	m	NOUN
ejpam-5974	185	13	(	(	PUNCT
ejpam-5974	185	14	n+	n+	NUM
ejpam-5974	185	15	1)(n+	1)(n+	NUM
ejpam-5974	185	16	1−	1−	NUM
ejpam-5974	185	17	1	1	NUM
ejpam-5974	185	18	)	)	PUNCT
ejpam-5974	185	19	=	=	SYM
ejpam-5974	186	1	2(2n−	2(2n−	NUM
ejpam-5974	186	2	1	1	NUM
ejpam-5974	186	3	)	)	PUNCT
ejpam-5974	186	4	n(n−	n(n−	NOUN
ejpam-5974	186	5	1	1	NUM
ejpam-5974	186	6	)	)	PUNCT
ejpam-5974	186	7	=	=	PUNCT
ejpam-5974	187	1	4n−	4n−	NUM
ejpam-5974	187	2	2	2	NUM
ejpam-5974	187	3	(	(	PUNCT
ejpam-5974	187	4	n+	n+	NUM
ejpam-5974	187	5	1)n	1)n	NUM
ejpam-5974	187	6	.	.	PUNCT
ejpam-5974	188	1	illustration	illustration	NOUN
ejpam-5974	188	2	:	:	PUNCT
ejpam-5974	188	3	consider	consider	VERB
ejpam-5974	188	4	figure	figure	NOUN
ejpam-5974	188	5	10	10	NUM
ejpam-5974	188	6	,	,	PUNCT
ejpam-5974	188	7	it	it	PRON
ejpam-5974	188	8	shows	show	VERB
ejpam-5974	188	9	that	that	SCONJ
ejpam-5974	188	10	the	the	DET
ejpam-5974	188	11	number	number	NOUN
ejpam-5974	188	12	of	of	ADP
ejpam-5974	188	13	edges	edge	NOUN
ejpam-5974	188	14	and	and	CCONJ
ejpam-5974	188	15	vertices	vertex	NOUN
ejpam-5974	188	16	of	of	ADP
ejpam-5974	188	17	f5	f5	NOUN
ejpam-5974	188	18	are	be	AUX
ejpam-5974	188	19	|e(f5)|	|e(f5)|	X
ejpam-5974	188	20	=	=	SYM
ejpam-5974	188	21	9	9	NUM
ejpam-5974	188	22	and	and	CCONJ
ejpam-5974	188	23	|v	|v	PROPN
ejpam-5974	188	24	(	(	PUNCT
ejpam-5974	188	25	f5)|	f5)|	PROPN
ejpam-5974	188	26	=	=	SYM
ejpam-5974	188	27	6	6	X
ejpam-5974	188	28	.	.	PUNCT
ejpam-5974	188	29	clearly	clearly	ADV
ejpam-5974	188	30	,	,	PUNCT
ejpam-5974	188	31	d(f5	d(f5	ADJ
ejpam-5974	188	32	)	)	PUNCT
ejpam-5974	188	33	=	=	SYM
ejpam-5974	189	1	2|e(f5)|	2|e(f5)|	PROPN
ejpam-5974	189	2	|v	|v	NOUN
ejpam-5974	189	3	(	(	PUNCT
ejpam-5974	189	4	f5)|(|v	f5)|(|v	PROPN
ejpam-5974	189	5	(	(	PUNCT
ejpam-5974	189	6	f5)|	f5)|	PROPN
ejpam-5974	189	7	−	−	PROPN
ejpam-5974	189	8	1	1	X
ejpam-5974	189	9	)	)	PUNCT
ejpam-5974	189	10	=	=	SYM
ejpam-5974	189	11	2|9|	2|9|	NUM
ejpam-5974	189	12	6(6−	6(6−	NUM
ejpam-5974	189	13	1	1	NUM
ejpam-5974	189	14	)	)	PUNCT
ejpam-5974	189	15	=	=	SYM
ejpam-5974	189	16	18	18	NUM
ejpam-5974	189	17	6(5	6(5	NUM
ejpam-5974	189	18	)	)	PUNCT
ejpam-5974	189	19	=	=	SYM
ejpam-5974	189	20	18	18	NUM
ejpam-5974	189	21	30	30	NUM
ejpam-5974	189	22	=	=	SYM
ejpam-5974	189	23	0.6	0.6	NUM
ejpam-5974	189	24	or	or	CCONJ
ejpam-5974	189	25	d(f5	d(f5	ADJ
ejpam-5974	189	26	)	)	PUNCT
ejpam-5974	189	27	=	=	PUNCT
ejpam-5974	190	1	4n−	4n−	NUM
ejpam-5974	190	2	2	2	NUM
ejpam-5974	190	3	(	(	PUNCT
ejpam-5974	190	4	n+	n+	X
ejpam-5974	190	5	1)n	1)n	NUM
ejpam-5974	190	6	=	=	SYM
ejpam-5974	190	7	4(5)−	4(5)−	SYM
ejpam-5974	190	8	2	2	NUM
ejpam-5974	190	9	(	(	PUNCT
ejpam-5974	190	10	5	5	NUM
ejpam-5974	190	11	+	+	NUM
ejpam-5974	190	12	1)5	1)5	NUM
ejpam-5974	190	13	=	=	SYM
ejpam-5974	190	14	18	18	NUM
ejpam-5974	190	15	(	(	PUNCT
ejpam-5974	190	16	5)5	5)5	NUM
ejpam-5974	190	17	=	=	SYM
ejpam-5974	190	18	18	18	NUM
ejpam-5974	190	19	28	28	NUM
ejpam-5974	190	20	=	=	SYM
ejpam-5974	190	21	0.6	0.6	NUM
ejpam-5974	190	22	.	.	PUNCT
ejpam-5974	190	23	1	1	NUM
ejpam-5974	190	24	2	2	NUM
ejpam-5974	190	25	3	3	NUM
ejpam-5974	190	26	4	4	NUM
ejpam-5974	190	27	5	5	NUM
ejpam-5974	190	28	v	v	NOUN
ejpam-5974	190	29	figure	figure	NOUN
ejpam-5974	190	30	10	10	NUM
ejpam-5974	190	31	:	:	PUNCT
ejpam-5974	190	32	the	the	DET
ejpam-5974	190	33	fan	fan	NOUN
ejpam-5974	190	34	graph	graph	NOUN
ejpam-5974	190	35	f5	f5	PROPN
ejpam-5974	190	36	r.	r.	PROPN
ejpam-5974	190	37	sango	sango	PROPN
ejpam-5974	190	38	,	,	PUNCT
ejpam-5974	190	39	i.	i.	NOUN
ejpam-5974	190	40	cabahug	cabahug	PROPN
ejpam-5974	190	41	,	,	PUNCT
ejpam-5974	190	42	jr	jr	PROPN
ejpam-5974	190	43	/	/	SYM
ejpam-5974	190	44	eur	eur	PROPN
ejpam-5974	190	45	.	.	PUNCT
ejpam-5974	191	1	j.	j.	PROPN
ejpam-5974	191	2	pure	pure	PROPN
ejpam-5974	191	3	appl	appl	PROPN
ejpam-5974	191	4	.	.	PROPN
ejpam-5974	191	5	math	math	PROPN
ejpam-5974	191	6	,	,	PUNCT
ejpam-5974	191	7	18	18	NUM
ejpam-5974	191	8	(	(	PUNCT
ejpam-5974	191	9	3	3	NUM
ejpam-5974	191	10	)	)	PUNCT
ejpam-5974	191	11	(	(	PUNCT
ejpam-5974	191	12	2025	2025	NUM
ejpam-5974	191	13	)	)	PUNCT
ejpam-5974	191	14	,	,	PUNCT
ejpam-5974	191	15	5974	5974	NUM
ejpam-5974	191	16	11	11	NUM
ejpam-5974	191	17	of	of	ADP
ejpam-5974	191	18	16	16	NUM
ejpam-5974	191	19	by	by	ADP
ejpam-5974	191	20	definition	definition	NOUN
ejpam-5974	191	21	mycielski	mycielski	NOUN
ejpam-5974	191	22	graph	graph	NOUN
ejpam-5974	191	23	,	,	PUNCT
ejpam-5974	191	24	we	we	PRON
ejpam-5974	191	25	can	can	AUX
ejpam-5974	191	26	easily	easily	ADV
ejpam-5974	191	27	derived	derive	VERB
ejpam-5974	191	28	the	the	DET
ejpam-5974	191	29	number	number	NOUN
ejpam-5974	191	30	of	of	ADP
ejpam-5974	191	31	edges	edge	NOUN
ejpam-5974	191	32	and	and	CCONJ
ejpam-5974	191	33	vertices	vertex	NOUN
ejpam-5974	191	34	of	of	ADP
ejpam-5974	191	35	a	a	DET
ejpam-5974	191	36	mycielski	mycielski	ADJ
ejpam-5974	191	37	graph	graph	NOUN
ejpam-5974	191	38	of	of	ADP
ejpam-5974	191	39	a	a	DET
ejpam-5974	191	40	connected	connected	ADJ
ejpam-5974	191	41	graph	graph	NOUN
ejpam-5974	191	42	.	.	PUNCT
ejpam-5974	192	1	remark	remark	PROPN
ejpam-5974	192	2	1	1	NUM
ejpam-5974	192	3	.	.	PUNCT
ejpam-5974	193	1	let	let	VERB
ejpam-5974	193	2	g	g	PRON
ejpam-5974	193	3	be	be	AUX
ejpam-5974	193	4	a	a	DET
ejpam-5974	193	5	nontrivial	nontrivial	ADJ
ejpam-5974	193	6	connected	connect	VERB
ejpam-5974	193	7	graph	graph	NOUN
ejpam-5974	193	8	.	.	PUNCT
ejpam-5974	194	1	then	then	ADV
ejpam-5974	194	2	|e(µ(g))|	|e(µ(g))|	PROPN
ejpam-5974	194	3	=	=	PUNCT
ejpam-5974	194	4	3|e(g)|+	3|e(g)|+	NUM
ejpam-5974	194	5	|v	|v	NOUN
ejpam-5974	194	6	(	(	PUNCT
ejpam-5974	194	7	g)|	g)|	NOUN
ejpam-5974	194	8	and	and	CCONJ
ejpam-5974	194	9	|v	|v	PROPN
ejpam-5974	194	10	(	(	PUNCT
ejpam-5974	194	11	µ(g))|	µ(g))|	X
ejpam-5974	194	12	=	=	SYM
ejpam-5974	194	13	2|v	2|v	PROPN
ejpam-5974	194	14	(	(	PUNCT
ejpam-5974	194	15	g)|+	g)|+	NOUN
ejpam-5974	194	16	1	1	NUM
ejpam-5974	194	17	.	.	PUNCT
ejpam-5974	194	18	theorem	theorem	NOUN
ejpam-5974	194	19	10	10	NUM
ejpam-5974	194	20	.	.	PUNCT
ejpam-5974	195	1	let	let	VERB
ejpam-5974	195	2	g	g	PRON
ejpam-5974	195	3	be	be	AUX
ejpam-5974	195	4	a	a	DET
ejpam-5974	195	5	nontrivial	nontrivial	ADJ
ejpam-5974	195	6	connected	connect	VERB
ejpam-5974	195	7	graph	graph	NOUN
ejpam-5974	195	8	.	.	PUNCT
ejpam-5974	196	1	then	then	ADV
ejpam-5974	196	2	the	the	DET
ejpam-5974	196	3	density	density	NOUN
ejpam-5974	196	4	of	of	ADP
ejpam-5974	196	5	µ(g	µ(g	PROPN
ejpam-5974	196	6	)	)	PUNCT
ejpam-5974	196	7	is	be	AUX
ejpam-5974	196	8	d(µ(g	d(µ(g	PROPN
ejpam-5974	196	9	)	)	PUNCT
ejpam-5974	196	10	)	)	PUNCT
ejpam-5974	197	1	=	=	PUNCT
ejpam-5974	198	1	3|e(g)|+	3|e(g)|+	NUM
ejpam-5974	198	2	|v	|v	NOUN
ejpam-5974	198	3	(	(	PUNCT
ejpam-5974	198	4	g)|	g)|	PROPN
ejpam-5974	198	5	|v	|v	PROPN
ejpam-5974	198	6	(	(	PUNCT
ejpam-5974	198	7	g)|(2|v	g)|(2|v	PROPN
ejpam-5974	198	8	(	(	PUNCT
ejpam-5974	198	9	g)|+	g)|+	NOUN
ejpam-5974	198	10	1	1	NUM
ejpam-5974	198	11	)	)	PUNCT
ejpam-5974	198	12	.	.	PUNCT
ejpam-5974	199	1	proof	proof	NOUN
ejpam-5974	199	2	.	.	PUNCT
ejpam-5974	200	1	by	by	ADP
ejpam-5974	200	2	definition	definition	NOUN
ejpam-5974	200	3	of	of	ADP
ejpam-5974	200	4	mycielski	mycielski	ADJ
ejpam-5974	200	5	graph	graph	NOUN
ejpam-5974	200	6	and	and	CCONJ
ejpam-5974	200	7	remark	remark	NOUN
ejpam-5974	200	8	1	1	NUM
ejpam-5974	200	9	,	,	PUNCT
ejpam-5974	200	10	d(µ(g	d(µ(g	PROPN
ejpam-5974	200	11	)	)	PUNCT
ejpam-5974	200	12	)	)	PUNCT
ejpam-5974	201	1	=	=	SYM
ejpam-5974	201	2	2|e(µ(g))|	2|e(µ(g))|	NUM
ejpam-5974	201	3	|v	|v	PROPN
ejpam-5974	201	4	(	(	PUNCT
ejpam-5974	201	5	g)|(|v	g)|(|v	X
ejpam-5974	201	6	(	(	PUNCT
ejpam-5974	201	7	g)|	g)|	NOUN
ejpam-5974	201	8	−	−	NOUN
ejpam-5974	201	9	1	1	NUM
ejpam-5974	201	10	)	)	PUNCT
ejpam-5974	201	11	=	=	SYM
ejpam-5974	201	12	2(3|e(g)|+	2(3|e(g)|+	NUM
ejpam-5974	201	13	|v	|v	X
ejpam-5974	201	14	(	(	PUNCT
ejpam-5974	201	15	g)|	g)|	PROPN
ejpam-5974	201	16	)	)	PUNCT
ejpam-5974	201	17	(	(	PUNCT
ejpam-5974	201	18	2|v	2|v	PROPN
ejpam-5974	201	19	(	(	PUNCT
ejpam-5974	201	20	g)|+	g)|+	PROPN
ejpam-5974	201	21	1)(2|v	1)(2|v	PROPN
ejpam-5974	201	22	(	(	PUNCT
ejpam-5974	201	23	g)|+	g)|+	PROPN
ejpam-5974	201	24	1−	1−	NUM
ejpam-5974	201	25	1	1	NUM
ejpam-5974	201	26	)	)	PUNCT
ejpam-5974	201	27	=	=	SYM
ejpam-5974	201	28	2(3|e(g)|+	2(3|e(g)|+	NUM
ejpam-5974	201	29	|v	|v	X
ejpam-5974	201	30	(	(	PUNCT
ejpam-5974	201	31	g)|	g)|	PROPN
ejpam-5974	201	32	)	)	PUNCT
ejpam-5974	201	33	(	(	PUNCT
ejpam-5974	201	34	2|v	2|v	PROPN
ejpam-5974	201	35	(	(	PUNCT
ejpam-5974	201	36	g)|+	g)|+	NOUN
ejpam-5974	201	37	1)2|v	1)2|v	NUM
ejpam-5974	201	38	(	(	PUNCT
ejpam-5974	201	39	g)|	g)|	NOUN
ejpam-5974	201	40	=	=	SYM
ejpam-5974	201	41	3|e(g)|+	3|e(g)|+	PROPN
ejpam-5974	201	42	|v	|v	NOUN
ejpam-5974	201	43	(	(	PUNCT
ejpam-5974	201	44	g)|	g)|	PROPN
ejpam-5974	201	45	|v	|v	PROPN
ejpam-5974	201	46	(	(	PUNCT
ejpam-5974	201	47	g)|(2|v	g)|(2|v	PROPN
ejpam-5974	201	48	(	(	PUNCT
ejpam-5974	201	49	g)|+	g)|+	NOUN
ejpam-5974	201	50	1	1	NUM
ejpam-5974	201	51	)	)	PUNCT
ejpam-5974	201	52	.	.	PUNCT
ejpam-5974	202	1	corollary	corollary	ADJ
ejpam-5974	202	2	1	1	NUM
ejpam-5974	202	3	.	.	PUNCT
ejpam-5974	203	1	let	let	VERB
ejpam-5974	203	2	g	g	PRON
ejpam-5974	203	3	be	be	AUX
ejpam-5974	203	4	a	a	DET
ejpam-5974	203	5	path	path	NOUN
ejpam-5974	203	6	graph	graph	NOUN
ejpam-5974	203	7	(	(	PUNCT
ejpam-5974	203	8	pn	pn	NOUN
ejpam-5974	203	9	)	)	PUNCT
ejpam-5974	203	10	where	where	SCONJ
ejpam-5974	203	11	n	n	PRON
ejpam-5974	203	12	≥	≥	NOUN
ejpam-5974	203	13	2	2	NUM
ejpam-5974	203	14	.	.	PUNCT
ejpam-5974	204	1	then	then	ADV
ejpam-5974	204	2	,	,	PUNCT
ejpam-5974	204	3	d(µ(pn	d(µ(pn	ADJ
ejpam-5974	204	4	)	)	PUNCT
ejpam-5974	204	5	)	)	PUNCT
ejpam-5974	205	1	=	=	PUNCT
ejpam-5974	206	1	4n−	4n−	NUM
ejpam-5974	206	2	3	3	NUM
ejpam-5974	206	3	2n2	2n2	NUM
ejpam-5974	206	4	+	+	CCONJ
ejpam-5974	206	5	n	n	NOUN
ejpam-5974	206	6	.	.	PUNCT
ejpam-5974	207	1	proof	proof	NOUN
ejpam-5974	207	2	.	.	PUNCT
ejpam-5974	208	1	by	by	ADP
ejpam-5974	208	2	theorem	theorem	NOUN
ejpam-5974	208	3	10	10	NUM
ejpam-5974	208	4	and	and	CCONJ
ejpam-5974	208	5	since	since	SCONJ
ejpam-5974	208	6	the	the	DET
ejpam-5974	208	7	number	number	NOUN
ejpam-5974	208	8	of	of	ADP
ejpam-5974	208	9	edges	edge	NOUN
ejpam-5974	208	10	and	and	CCONJ
ejpam-5974	208	11	vertices	vertex	NOUN
ejpam-5974	208	12	of	of	ADP
ejpam-5974	208	13	pn	pn	PROPN
ejpam-5974	208	14	are	be	AUX
ejpam-5974	208	15	n−	n−	NOUN
ejpam-5974	208	16	1	1	NUM
ejpam-5974	208	17	and	and	CCONJ
ejpam-5974	208	18	n	n	PRON
ejpam-5974	208	19	respectively	respectively	ADV
ejpam-5974	208	20	,	,	PUNCT
ejpam-5974	208	21	we	we	PRON
ejpam-5974	208	22	have	have	VERB
ejpam-5974	208	23	d(µ(pn	d(µ(pn	NOUN
ejpam-5974	208	24	)	)	PUNCT
ejpam-5974	208	25	)	)	PUNCT
ejpam-5974	209	1	=	=	SYM
ejpam-5974	209	2	3|e(pn)|+	3|e(pn)|+	NUM
ejpam-5974	209	3	|v	|v	NOUN
ejpam-5974	209	4	(	(	PUNCT
ejpam-5974	209	5	pn)|	pn)|	PROPN
ejpam-5974	209	6	|v	|v	X
ejpam-5974	209	7	(	(	PUNCT
ejpam-5974	209	8	pn)|(2|v	pn)|(2|v	PROPN
ejpam-5974	209	9	(	(	PUNCT
ejpam-5974	209	10	pn)|+	pn)|+	NOUN
ejpam-5974	209	11	1	1	NUM
ejpam-5974	209	12	)	)	PUNCT
ejpam-5974	209	13	=	=	PUNCT
ejpam-5974	209	14	3(n−	3(n−	NUM
ejpam-5974	209	15	1	1	NUM
ejpam-5974	209	16	)	)	PUNCT
ejpam-5974	209	17	+	+	CCONJ
ejpam-5974	209	18	n	n	CCONJ
ejpam-5974	209	19	n(2n+	n(2n+	NOUN
ejpam-5974	209	20	1	1	NUM
ejpam-5974	209	21	)	)	PUNCT
ejpam-5974	209	22	=	=	NOUN
ejpam-5974	210	1	3n−	3n−	NUM
ejpam-5974	210	2	3	3	NUM
ejpam-5974	210	3	+	+	NUM
ejpam-5974	210	4	n	n	PRON
ejpam-5974	210	5	2n2	2n2	NUM
ejpam-5974	210	6	+	+	CCONJ
ejpam-5974	210	7	n	n	NOUN
ejpam-5974	210	8	=	=	SYM
ejpam-5974	210	9	4n−	4n−	NUM
ejpam-5974	210	10	3	3	NUM
ejpam-5974	210	11	2n2	2n2	NUM
ejpam-5974	210	12	+	+	CCONJ
ejpam-5974	210	13	n	n	NOUN
ejpam-5974	210	14	.	.	PUNCT
ejpam-5974	211	1	corollary	corollary	ADJ
ejpam-5974	211	2	2	2	NUM
ejpam-5974	211	3	.	.	PUNCT
ejpam-5974	212	1	let	let	VERB
ejpam-5974	212	2	g	g	PRON
ejpam-5974	212	3	be	be	AUX
ejpam-5974	212	4	a	a	DET
ejpam-5974	212	5	cycle	cycle	NOUN
ejpam-5974	212	6	graph	graph	NOUN
ejpam-5974	212	7	(	(	PUNCT
ejpam-5974	212	8	cn	cn	PROPN
ejpam-5974	212	9	)	)	PUNCT
ejpam-5974	212	10	with	with	ADP
ejpam-5974	212	11	n	n	PRON
ejpam-5974	212	12	≥	≥	NUM
ejpam-5974	212	13	3	3	NUM
ejpam-5974	212	14	.	.	PUNCT
ejpam-5974	213	1	then	then	ADV
ejpam-5974	213	2	d(µ(cn	d(µ(cn	NOUN
ejpam-5974	213	3	)	)	PUNCT
ejpam-5974	213	4	)	)	PUNCT
ejpam-5974	214	1	=	=	PUNCT
ejpam-5974	214	2	4	4	NUM
ejpam-5974	214	3	2n+	2n+	NUM
ejpam-5974	214	4	1	1	NUM
ejpam-5974	214	5	.	.	PUNCT
ejpam-5974	215	1	proof	proof	NOUN
ejpam-5974	215	2	.	.	PUNCT
ejpam-5974	216	1	by	by	ADP
ejpam-5974	216	2	theorem	theorem	NOUN
ejpam-5974	216	3	10	10	NUM
ejpam-5974	216	4	and	and	CCONJ
ejpam-5974	216	5	since	since	SCONJ
ejpam-5974	216	6	the	the	DET
ejpam-5974	216	7	number	number	NOUN
ejpam-5974	216	8	of	of	ADP
ejpam-5974	216	9	edges	edge	NOUN
ejpam-5974	216	10	and	and	CCONJ
ejpam-5974	216	11	vertices	vertex	NOUN
ejpam-5974	216	12	of	of	ADP
ejpam-5974	216	13	cn	cn	PROPN
ejpam-5974	216	14	is	be	AUX
ejpam-5974	216	15	n	n	CCONJ
ejpam-5974	216	16	,	,	PUNCT
ejpam-5974	216	17	we	we	PRON
ejpam-5974	216	18	have	have	VERB
ejpam-5974	216	19	d(µ(cn	d(µ(cn	NOUN
ejpam-5974	216	20	)	)	PUNCT
ejpam-5974	216	21	)	)	PUNCT
ejpam-5974	217	1	=	=	SYM
ejpam-5974	217	2	3|e(cn)|+	3|e(cn)|+	NUM
ejpam-5974	217	3	|v	|v	X
ejpam-5974	217	4	(	(	PUNCT
ejpam-5974	217	5	cn)|	cn)|	PROPN
ejpam-5974	217	6	|v	|v	PROPN
ejpam-5974	217	7	(	(	PUNCT
ejpam-5974	217	8	cn)|(2|v	cn)|(2|v	PROPN
ejpam-5974	217	9	(	(	PUNCT
ejpam-5974	217	10	cn)|+	cn)|+	PROPN
ejpam-5974	217	11	1	1	NUM
ejpam-5974	217	12	)	)	PUNCT
ejpam-5974	217	13	=	=	SYM
ejpam-5974	217	14	3n+	3n+	NUM
ejpam-5974	217	15	n	n	PROPN
ejpam-5974	217	16	n(2n+	n(2n+	NOUN
ejpam-5974	217	17	1	1	NUM
ejpam-5974	217	18	)	)	PUNCT
ejpam-5974	217	19	=	=	NOUN
ejpam-5974	217	20	4n	4n	ADJ
ejpam-5974	217	21	n(2n+	n(2n+	NOUN
ejpam-5974	217	22	1	1	NUM
ejpam-5974	217	23	)	)	PUNCT
ejpam-5974	217	24	=	=	SYM
ejpam-5974	217	25	4	4	NUM
ejpam-5974	217	26	2n+	2n+	NUM
ejpam-5974	217	27	1	1	NUM
ejpam-5974	217	28	.	.	PUNCT
ejpam-5974	218	1	corollary	corollary	ADJ
ejpam-5974	218	2	3	3	X
ejpam-5974	218	3	.	.	PUNCT
ejpam-5974	219	1	let	let	VERB
ejpam-5974	219	2	g	g	PRON
ejpam-5974	219	3	be	be	AUX
ejpam-5974	219	4	a	a	DET
ejpam-5974	219	5	complete	complete	ADJ
ejpam-5974	219	6	graph	graph	NOUN
ejpam-5974	219	7	(	(	PUNCT
ejpam-5974	219	8	kn	kn	PROPN
ejpam-5974	219	9	)	)	PUNCT
ejpam-5974	219	10	where	where	SCONJ
ejpam-5974	219	11	n	n	PRON
ejpam-5974	219	12	≥	≥	NOUN
ejpam-5974	219	13	4	4	NUM
ejpam-5974	219	14	.	.	PUNCT
ejpam-5974	219	15	then	then	ADV
ejpam-5974	219	16	d(µ(kn	d(µ(kn	NOUN
ejpam-5974	219	17	)	)	PUNCT
ejpam-5974	219	18	)	)	PUNCT
ejpam-5974	220	1	=	=	PUNCT
ejpam-5974	221	1	3n−	3n−	NUM
ejpam-5974	221	2	1	1	NUM
ejpam-5974	221	3	4n+	4n+	NUM
ejpam-5974	221	4	2	2	NUM
ejpam-5974	221	5	.	.	PUNCT
ejpam-5974	222	1	r.	r.	PROPN
ejpam-5974	222	2	sango	sango	PROPN
ejpam-5974	222	3	,	,	PUNCT
ejpam-5974	222	4	i.	i.	NOUN
ejpam-5974	222	5	cabahug	cabahug	PROPN
ejpam-5974	222	6	,	,	PUNCT
ejpam-5974	222	7	jr	jr	PROPN
ejpam-5974	222	8	/	/	SYM
ejpam-5974	222	9	eur	eur	PROPN
ejpam-5974	222	10	.	.	PUNCT
ejpam-5974	223	1	j.	j.	PROPN
ejpam-5974	223	2	pure	pure	PROPN
ejpam-5974	223	3	appl	appl	PROPN
ejpam-5974	223	4	.	.	PROPN
ejpam-5974	223	5	math	math	PROPN
ejpam-5974	223	6	,	,	PUNCT
ejpam-5974	223	7	18	18	NUM
ejpam-5974	223	8	(	(	PUNCT
ejpam-5974	223	9	3	3	NUM
ejpam-5974	223	10	)	)	PUNCT
ejpam-5974	223	11	(	(	PUNCT
ejpam-5974	223	12	2025	2025	NUM
ejpam-5974	223	13	)	)	PUNCT
ejpam-5974	223	14	,	,	PUNCT
ejpam-5974	223	15	5974	5974	NUM
ejpam-5974	223	16	12	12	NUM
ejpam-5974	223	17	of	of	ADP
ejpam-5974	223	18	16	16	NUM
ejpam-5974	223	19	proof	proof	NOUN
ejpam-5974	223	20	.	.	PUNCT
ejpam-5974	224	1	by	by	ADP
ejpam-5974	224	2	theorem	theorem	NOUN
ejpam-5974	224	3	10	10	NUM
ejpam-5974	224	4	and	and	CCONJ
ejpam-5974	224	5	since	since	SCONJ
ejpam-5974	224	6	the	the	DET
ejpam-5974	224	7	number	number	NOUN
ejpam-5974	224	8	of	of	ADP
ejpam-5974	224	9	edges	edge	NOUN
ejpam-5974	224	10	and	and	CCONJ
ejpam-5974	224	11	vertices	vertex	NOUN
ejpam-5974	224	12	of	of	ADP
ejpam-5974	224	13	kn	kn	PROPN
ejpam-5974	224	14	are	be	AUX
ejpam-5974	224	15	n(n−	n(n−	PROPN
ejpam-5974	224	16	1	1	NUM
ejpam-5974	224	17	)	)	PUNCT
ejpam-5974	224	18	2	2	NUM
ejpam-5974	224	19	and	and	CCONJ
ejpam-5974	224	20	n	n	PRON
ejpam-5974	224	21	respectively	respectively	ADV
ejpam-5974	224	22	,	,	PUNCT
ejpam-5974	224	23	we	we	PRON
ejpam-5974	224	24	have	have	VERB
ejpam-5974	224	25	d(µ(kn	d(µ(kn	NOUN
ejpam-5974	224	26	)	)	PUNCT
ejpam-5974	224	27	)	)	PUNCT
ejpam-5974	225	1	=	=	SYM
ejpam-5974	226	1	3|e(kn)|+	3|e(kn)|+	NUM
ejpam-5974	226	2	|v	|v	NOUN
ejpam-5974	226	3	(	(	PUNCT
ejpam-5974	226	4	kn)|	kn)|	PROPN
ejpam-5974	226	5	|v	|v	PROPN
ejpam-5974	226	6	(	(	PUNCT
ejpam-5974	226	7	kn)|(2|v	kn)|(2|v	PROPN
ejpam-5974	226	8	(	(	PUNCT
ejpam-5974	226	9	kn)|+	kn)|+	PROPN
ejpam-5974	226	10	1	1	NUM
ejpam-5974	226	11	)	)	PUNCT
ejpam-5974	226	12	=	=	NOUN
ejpam-5974	227	1	3(n2	3(n2	NUM
ejpam-5974	227	2	−	−	NOUN
ejpam-5974	227	3	n	n	CCONJ
ejpam-5974	227	4	)	)	PUNCT
ejpam-5974	227	5	2	2	NUM
ejpam-5974	227	6	+	+	SYM
ejpam-5974	227	7	n	n	PRON
ejpam-5974	227	8	2n2	2n2	NUM
ejpam-5974	227	9	+	+	CCONJ
ejpam-5974	227	10	n	n	NOUN
ejpam-5974	227	11	=	=	SYM
ejpam-5974	227	12	3n2	3n2	NUM
ejpam-5974	227	13	−	−	NUM
ejpam-5974	227	14	3n+	3n+	NUM
ejpam-5974	227	15	2n	2n	NUM
ejpam-5974	227	16	2	2	NUM
ejpam-5974	227	17	2n2	2n2	NUM
ejpam-5974	227	18	+	+	CCONJ
ejpam-5974	227	19	n	n	NOUN
ejpam-5974	227	20	=	=	SYM
ejpam-5974	227	21	3n2	3n2	NUM
ejpam-5974	227	22	−	−	NUM
ejpam-5974	227	23	n	n	CCONJ
ejpam-5974	227	24	2	2	NUM
ejpam-5974	227	25	2n2	2n2	NUM
ejpam-5974	227	26	+	+	CCONJ
ejpam-5974	227	27	n	n	NOUN
ejpam-5974	227	28	=	=	SYM
ejpam-5974	227	29	(	(	PUNCT
ejpam-5974	227	30	3n2	3n2	NUM
ejpam-5974	227	31	−	−	PROPN
ejpam-5974	227	32	n	n	PRON
ejpam-5974	227	33	2	2	NUM
ejpam-5974	227	34	)	)	PUNCT
ejpam-5974	227	35	(	(	PUNCT
ejpam-5974	227	36	1	1	NUM
ejpam-5974	227	37	2n2	2n2	NUM
ejpam-5974	227	38	+	+	CCONJ
ejpam-5974	227	39	n	n	CCONJ
ejpam-5974	227	40	)	)	PUNCT
ejpam-5974	228	1	=	=	SYM
ejpam-5974	228	2	3n2	3n2	NUM
ejpam-5974	228	3	−	−	NUM
ejpam-5974	229	1	n	n	PRON
ejpam-5974	229	2	4n2	4n2	NUM
ejpam-5974	230	1	+	+	NUM
ejpam-5974	230	2	2n	2n	NUM
ejpam-5974	230	3	=	=	SYM
ejpam-5974	230	4	n(3n−	n(3n−	PROPN
ejpam-5974	230	5	1	1	NUM
ejpam-5974	230	6	)	)	PUNCT
ejpam-5974	230	7	n(4n+	n(4n+	NOUN
ejpam-5974	230	8	2	2	NUM
ejpam-5974	230	9	)	)	PUNCT
ejpam-5974	230	10	=	=	SYM
ejpam-5974	231	1	3n−	3n−	NUM
ejpam-5974	231	2	1	1	NUM
ejpam-5974	231	3	4n+	4n+	NUM
ejpam-5974	231	4	2	2	NUM
ejpam-5974	231	5	.	.	PUNCT
ejpam-5974	232	1	corollary	corollary	ADJ
ejpam-5974	232	2	4	4	NUM
ejpam-5974	232	3	.	.	PUNCT
ejpam-5974	233	1	let	let	VERB
ejpam-5974	233	2	g	g	PRON
ejpam-5974	233	3	be	be	AUX
ejpam-5974	233	4	a	a	DET
ejpam-5974	233	5	barbell	barbell	NOUN
ejpam-5974	233	6	graph	graph	NOUN
ejpam-5974	233	7	(	(	PUNCT
ejpam-5974	233	8	bn	bn	NOUN
ejpam-5974	233	9	)	)	PUNCT
ejpam-5974	233	10	where	where	SCONJ
ejpam-5974	233	11	n	n	PRON
ejpam-5974	233	12	≥	≥	NOUN
ejpam-5974	233	13	3	3	NUM
ejpam-5974	233	14	.	.	PUNCT
ejpam-5974	234	1	then	then	ADV
ejpam-5974	234	2	d(µ(bn	d(µ(bn	NOUN
ejpam-5974	234	3	)	)	PUNCT
ejpam-5974	234	4	)	)	PUNCT
ejpam-5974	235	1	=	=	SYM
ejpam-5974	235	2	3n2	3n2	NUM
ejpam-5974	235	3	−	−	NUM
ejpam-5974	235	4	n+	n+	ADP
ejpam-5974	235	5	3	3	NUM
ejpam-5974	235	6	8n2	8n2	NUM
ejpam-5974	235	7	+	+	CCONJ
ejpam-5974	235	8	2n	2n	NUM
ejpam-5974	235	9	.	.	PUNCT
ejpam-5974	236	1	proof	proof	NOUN
ejpam-5974	236	2	.	.	PUNCT
ejpam-5974	237	1	by	by	ADP
ejpam-5974	237	2	theorem	theorem	NOUN
ejpam-5974	237	3	10	10	NUM
ejpam-5974	237	4	and	and	CCONJ
ejpam-5974	237	5	since	since	SCONJ
ejpam-5974	237	6	the	the	DET
ejpam-5974	237	7	number	number	NOUN
ejpam-5974	237	8	of	of	ADP
ejpam-5974	237	9	edges	edge	NOUN
ejpam-5974	237	10	and	and	CCONJ
ejpam-5974	237	11	vertices	vertex	NOUN
ejpam-5974	237	12	of	of	ADP
ejpam-5974	237	13	bn	bn	NOUN
ejpam-5974	237	14	are	be	AUX
ejpam-5974	237	15	n2−n+1	n2−n+1	NUM
ejpam-5974	237	16	and	and	CCONJ
ejpam-5974	237	17	2n	2n	NUM
ejpam-5974	237	18	respectively	respectively	ADV
ejpam-5974	237	19	,	,	PUNCT
ejpam-5974	237	20	we	we	PRON
ejpam-5974	237	21	have	have	VERB
ejpam-5974	237	22	d(µ(bn	d(µ(bn	NOUN
ejpam-5974	237	23	)	)	PUNCT
ejpam-5974	237	24	)	)	PUNCT
ejpam-5974	238	1	=	=	PUNCT
ejpam-5974	238	2	3|e(bn)|+	3|e(bn)|+	NUM
ejpam-5974	238	3	|v	|v	NOUN
ejpam-5974	238	4	(	(	PUNCT
ejpam-5974	238	5	bn)|	bn)|	PROPN
ejpam-5974	238	6	|v	|v	PROPN
ejpam-5974	238	7	(	(	PUNCT
ejpam-5974	238	8	bn)|(2|v	bn)|(2|v	X
ejpam-5974	238	9	(	(	PUNCT
ejpam-5974	238	10	bn)|+	bn)|+	ADJ
ejpam-5974	238	11	1	1	NUM
ejpam-5974	238	12	)	)	PUNCT
ejpam-5974	238	13	=	=	NOUN
ejpam-5974	239	1	3(n2	3(n2	NUM
ejpam-5974	239	2	−	−	NUM
ejpam-5974	239	3	n+	n+	ADP
ejpam-5974	239	4	1	1	NUM
ejpam-5974	239	5	)	)	PUNCT
ejpam-5974	239	6	+	+	NUM
ejpam-5974	239	7	2n	2n	NUM
ejpam-5974	239	8	2n(2	2n(2	NUM
ejpam-5974	239	9	·	·	SYM
ejpam-5974	239	10	2n+	2n+	NUM
ejpam-5974	239	11	1	1	NUM
ejpam-5974	239	12	)	)	PUNCT
ejpam-5974	239	13	=	=	SYM
ejpam-5974	240	1	3n2	3n2	NUM
ejpam-5974	240	2	−	−	NUM
ejpam-5974	240	3	3n+	3n+	NUM
ejpam-5974	240	4	3	3	NUM
ejpam-5974	240	5	+	+	NUM
ejpam-5974	240	6	2n	2n	NUM
ejpam-5974	240	7	8n2	8n2	NUM
ejpam-5974	241	1	+	+	CCONJ
ejpam-5974	241	2	2n	2n	NUM
ejpam-5974	241	3	=	=	SYM
ejpam-5974	242	1	3n2	3n2	NUM
ejpam-5974	242	2	−	−	NUM
ejpam-5974	242	3	n+	n+	ADP
ejpam-5974	242	4	3	3	NUM
ejpam-5974	242	5	8n2	8n2	NUM
ejpam-5974	242	6	+	+	CCONJ
ejpam-5974	242	7	2n	2n	NUM
ejpam-5974	242	8	.	.	PUNCT
ejpam-5974	243	1	corollary	corollary	ADJ
ejpam-5974	243	2	5	5	NUM
ejpam-5974	243	3	.	.	PUNCT
ejpam-5974	244	1	let	let	VERB
ejpam-5974	244	2	g	g	PRON
ejpam-5974	244	3	be	be	AUX
ejpam-5974	244	4	a	a	DET
ejpam-5974	244	5	friendship	friendship	NOUN
ejpam-5974	244	6	graph	graph	NOUN
ejpam-5974	244	7	(	(	PUNCT
ejpam-5974	244	8	frn	frn	PROPN
ejpam-5974	244	9	)	)	PUNCT
ejpam-5974	244	10	where	where	SCONJ
ejpam-5974	244	11	n	n	PRON
ejpam-5974	244	12	≥	≥	NOUN
ejpam-5974	244	13	2	2	NUM
ejpam-5974	244	14	.	.	PUNCT
ejpam-5974	245	1	then	then	ADV
ejpam-5974	245	2	d(µ(frn	d(µ(frn	NUM
ejpam-5974	245	3	)	)	PUNCT
ejpam-5974	245	4	)	)	PUNCT
ejpam-5974	246	1	=	=	PUNCT
ejpam-5974	246	2	11n+	11n+	NUM
ejpam-5974	246	3	1	1	NUM
ejpam-5974	246	4	8n2	8n2	NUM
ejpam-5974	246	5	+	+	CCONJ
ejpam-5974	246	6	10n+	10n+	NUM
ejpam-5974	246	7	3	3	NUM
ejpam-5974	246	8	.	.	PUNCT
ejpam-5974	247	1	proof	proof	NOUN
ejpam-5974	247	2	.	.	PUNCT
ejpam-5974	248	1	by	by	ADP
ejpam-5974	248	2	theorem	theorem	NOUN
ejpam-5974	248	3	10	10	NUM
ejpam-5974	248	4	and	and	CCONJ
ejpam-5974	248	5	since	since	SCONJ
ejpam-5974	248	6	the	the	DET
ejpam-5974	248	7	number	number	NOUN
ejpam-5974	248	8	of	of	ADP
ejpam-5974	248	9	edges	edge	NOUN
ejpam-5974	248	10	and	and	CCONJ
ejpam-5974	248	11	vertices	vertex	NOUN
ejpam-5974	248	12	of	of	ADP
ejpam-5974	248	13	frn	frn	PROPN
ejpam-5974	248	14	are	be	AUX
ejpam-5974	248	15	3n	3n	NUM
ejpam-5974	248	16	and	and	CCONJ
ejpam-5974	248	17	3n−	3n−	PROPN
ejpam-5974	248	18	2	2	NUM
ejpam-5974	248	19	respectively	respectively	ADV
ejpam-5974	248	20	,	,	PUNCT
ejpam-5974	248	21	we	we	PRON
ejpam-5974	248	22	have	have	VERB
ejpam-5974	248	23	d(µ(frn	d(µ(frn	NUM
ejpam-5974	248	24	)	)	PUNCT
ejpam-5974	248	25	)	)	PUNCT
ejpam-5974	249	1	=	=	PUNCT
ejpam-5974	250	1	3|e(frn)|+	3|e(frn)|+	NUM
ejpam-5974	250	2	|v	|v	NOUN
ejpam-5974	250	3	(	(	PUNCT
ejpam-5974	250	4	frn)|	frn)|	PROPN
ejpam-5974	250	5	|v	|v	PROPN
ejpam-5974	250	6	(	(	PUNCT
ejpam-5974	250	7	frn)|(2|v	frn)|(2|v	PROPN
ejpam-5974	250	8	(	(	PUNCT
ejpam-5974	250	9	frn)|+	frn)|+	NOUN
ejpam-5974	250	10	1	1	NUM
ejpam-5974	250	11	)	)	PUNCT
ejpam-5974	250	12	=	=	SYM
ejpam-5974	250	13	3(3n	3(3n	NUM
ejpam-5974	250	14	)	)	PUNCT
ejpam-5974	250	15	+	+	CCONJ
ejpam-5974	250	16	2n+	2n+	NUM
ejpam-5974	250	17	1	1	NUM
ejpam-5974	250	18	(	(	PUNCT
ejpam-5974	250	19	2n+	2n+	NUM
ejpam-5974	250	20	1)[2(2n+	1)[2(2n+	NUM
ejpam-5974	250	21	1	1	NUM
ejpam-5974	250	22	)	)	PUNCT
ejpam-5974	250	23	+	+	CCONJ
ejpam-5974	250	24	1	1	X
ejpam-5974	250	25	]	]	X
ejpam-5974	250	26	=	=	SYM
ejpam-5974	250	27	9n+	9n+	NUM
ejpam-5974	250	28	2n+	2n+	NUM
ejpam-5974	250	29	1	1	NUM
ejpam-5974	250	30	(	(	PUNCT
ejpam-5974	250	31	2n+	2n+	NUM
ejpam-5974	250	32	1)(4n+	1)(4n+	NOUN
ejpam-5974	250	33	3	3	NUM
ejpam-5974	250	34	)	)	PUNCT
ejpam-5974	250	35	=	=	PUNCT
ejpam-5974	250	36	11n+	11n+	NUM
ejpam-5974	250	37	1	1	NUM
ejpam-5974	250	38	8n2	8n2	NUM
ejpam-5974	250	39	+	+	CCONJ
ejpam-5974	250	40	10n+	10n+	NUM
ejpam-5974	250	41	3	3	NUM
ejpam-5974	250	42	.	.	PUNCT
ejpam-5974	250	43	corollary	corollary	ADJ
ejpam-5974	250	44	6	6	NUM
ejpam-5974	250	45	.	.	PUNCT
ejpam-5974	251	1	let	let	VERB
ejpam-5974	251	2	g	g	PRON
ejpam-5974	251	3	be	be	AUX
ejpam-5974	251	4	a	a	DET
ejpam-5974	251	5	sunlet	sunlet	NOUN
ejpam-5974	251	6	graph	graph	NOUN
ejpam-5974	251	7	(	(	PUNCT
ejpam-5974	251	8	sn	sn	PROPN
ejpam-5974	251	9	)	)	PUNCT
ejpam-5974	251	10	where	where	SCONJ
ejpam-5974	251	11	n	n	PRON
ejpam-5974	251	12	≥	≥	NOUN
ejpam-5974	251	13	3	3	NUM
ejpam-5974	251	14	.	.	PUNCT
ejpam-5974	251	15	then	then	ADV
ejpam-5974	251	16	d(µ(sn	d(µ(sn	NOUN
ejpam-5974	251	17	)	)	PUNCT
ejpam-5974	251	18	)	)	PUNCT
ejpam-5974	252	1	=	=	PUNCT
ejpam-5974	252	2	4	4	NUM
ejpam-5974	252	3	4n+	4n+	NUM
ejpam-5974	252	4	1	1	NUM
ejpam-5974	252	5	.	.	PUNCT
ejpam-5974	253	1	r.	r.	PROPN
ejpam-5974	253	2	sango	sango	PROPN
ejpam-5974	253	3	,	,	PUNCT
ejpam-5974	253	4	i.	i.	NOUN
ejpam-5974	253	5	cabahug	cabahug	PROPN
ejpam-5974	253	6	,	,	PUNCT
ejpam-5974	253	7	jr	jr	PROPN
ejpam-5974	253	8	/	/	SYM
ejpam-5974	253	9	eur	eur	PROPN
ejpam-5974	253	10	.	.	PUNCT
ejpam-5974	254	1	j.	j.	PROPN
ejpam-5974	254	2	pure	pure	PROPN
ejpam-5974	254	3	appl	appl	PROPN
ejpam-5974	254	4	.	.	PROPN
ejpam-5974	254	5	math	math	PROPN
ejpam-5974	254	6	,	,	PUNCT
ejpam-5974	254	7	18	18	NUM
ejpam-5974	254	8	(	(	PUNCT
ejpam-5974	254	9	3	3	NUM
ejpam-5974	254	10	)	)	PUNCT
ejpam-5974	254	11	(	(	PUNCT
ejpam-5974	254	12	2025	2025	NUM
ejpam-5974	254	13	)	)	PUNCT
ejpam-5974	254	14	,	,	PUNCT
ejpam-5974	254	15	5974	5974	NUM
ejpam-5974	254	16	13	13	NUM
ejpam-5974	254	17	of	of	ADP
ejpam-5974	254	18	16	16	NUM
ejpam-5974	254	19	proof	proof	NOUN
ejpam-5974	254	20	.	.	PUNCT
ejpam-5974	255	1	by	by	ADP
ejpam-5974	255	2	theorem	theorem	NOUN
ejpam-5974	255	3	10	10	NUM
ejpam-5974	255	4	and	and	CCONJ
ejpam-5974	255	5	since	since	SCONJ
ejpam-5974	255	6	the	the	DET
ejpam-5974	255	7	number	number	NOUN
ejpam-5974	255	8	of	of	ADP
ejpam-5974	255	9	edges	edge	NOUN
ejpam-5974	255	10	and	and	CCONJ
ejpam-5974	255	11	vertices	vertex	NOUN
ejpam-5974	255	12	of	of	ADP
ejpam-5974	255	13	sn	sn	PROPN
ejpam-5974	255	14	is	be	AUX
ejpam-5974	255	15	2n	2n	NUM
ejpam-5974	255	16	,	,	PUNCT
ejpam-5974	255	17	we	we	PRON
ejpam-5974	255	18	have	have	AUX
ejpam-5974	255	19	d(µ(sn	d(µ(sn	VERB
ejpam-5974	255	20	)	)	PUNCT
ejpam-5974	255	21	)	)	PUNCT
ejpam-5974	256	1	=	=	PUNCT
ejpam-5974	257	1	3|e(sn)|+	3|e(sn)|+	NUM
ejpam-5974	257	2	|v	|v	NOUN
ejpam-5974	257	3	(	(	PUNCT
ejpam-5974	257	4	sn)|	sn)|	NOUN
ejpam-5974	257	5	|v	|v	NOUN
ejpam-5974	257	6	(	(	PUNCT
ejpam-5974	257	7	sn)|(2|v	sn)|(2|v	NOUN
ejpam-5974	257	8	(	(	PUNCT
ejpam-5974	257	9	sn)|+	sn)|+	PROPN
ejpam-5974	257	10	1	1	NUM
ejpam-5974	257	11	)	)	PUNCT
ejpam-5974	257	12	=	=	SYM
ejpam-5974	257	13	3(2n	3(2n	NUM
ejpam-5974	257	14	)	)	PUNCT
ejpam-5974	257	15	+	+	NUM
ejpam-5974	257	16	2n	2n	NUM
ejpam-5974	257	17	2n(2	2n(2	NUM
ejpam-5974	257	18	·	·	SYM
ejpam-5974	257	19	2n+	2n+	NUM
ejpam-5974	257	20	1	1	NUM
ejpam-5974	257	21	)	)	PUNCT
ejpam-5974	257	22	=	=	SYM
ejpam-5974	257	23	2n(4	2n(4	NUM
ejpam-5974	257	24	)	)	PUNCT
ejpam-5974	257	25	2n(4n+	2n(4n+	NUM
ejpam-5974	257	26	1	1	NUM
ejpam-5974	257	27	)	)	PUNCT
ejpam-5974	257	28	=	=	SYM
ejpam-5974	257	29	4	4	NUM
ejpam-5974	257	30	4n+	4n+	NUM
ejpam-5974	257	31	1	1	NUM
ejpam-5974	257	32	.	.	PUNCT
ejpam-5974	258	1	corollary	corollary	ADJ
ejpam-5974	258	2	7	7	NUM
ejpam-5974	258	3	.	.	PUNCT
ejpam-5974	259	1	let	let	VERB
ejpam-5974	259	2	g	g	PRON
ejpam-5974	259	3	be	be	AUX
ejpam-5974	259	4	a	a	DET
ejpam-5974	259	5	banana	banana	NOUN
ejpam-5974	259	6	graph	graph	NOUN
ejpam-5974	259	7	(	(	PUNCT
ejpam-5974	259	8	bn	bn	INTJ
ejpam-5974	259	9	,	,	PUNCT
ejpam-5974	259	10	k	k	NOUN
ejpam-5974	259	11	)	)	PUNCT
ejpam-5974	259	12	where	where	SCONJ
ejpam-5974	259	13	n	n	PRON
ejpam-5974	259	14	≥	≥	X
ejpam-5974	259	15	2	2	NUM
ejpam-5974	259	16	and	and	CCONJ
ejpam-5974	259	17	k	k	PROPN
ejpam-5974	259	18	≥	≥	NUM
ejpam-5974	259	19	4	4	NUM
ejpam-5974	259	20	.	.	PUNCT
ejpam-5974	260	1	then	then	ADV
ejpam-5974	260	2	d(µ(bn	d(µ(bn	PROPN
ejpam-5974	260	3	,	,	PUNCT
ejpam-5974	260	4	k	k	NOUN
ejpam-5974	260	5	)	)	PUNCT
ejpam-5974	260	6	)	)	PUNCT
ejpam-5974	261	1	=	=	PUNCT
ejpam-5974	261	2	4nk	4nk	NOUN
ejpam-5974	262	1	+	+	CCONJ
ejpam-5974	262	2	1	1	NUM
ejpam-5974	262	3	2n2k2	2n2k2	NUM
ejpam-5974	262	4	+	+	CCONJ
ejpam-5974	262	5	5nk	5nk	ADJ
ejpam-5974	262	6	+	+	CCONJ
ejpam-5974	262	7	3	3	NUM
ejpam-5974	262	8	.	.	PUNCT
ejpam-5974	263	1	proof	proof	NOUN
ejpam-5974	263	2	.	.	PUNCT
ejpam-5974	264	1	by	by	ADP
ejpam-5974	264	2	theorem	theorem	NOUN
ejpam-5974	264	3	10	10	NUM
ejpam-5974	264	4	and	and	CCONJ
ejpam-5974	264	5	since	since	SCONJ
ejpam-5974	264	6	the	the	DET
ejpam-5974	264	7	number	number	NOUN
ejpam-5974	264	8	of	of	ADP
ejpam-5974	264	9	edges	edge	NOUN
ejpam-5974	264	10	and	and	CCONJ
ejpam-5974	264	11	vertices	vertex	NOUN
ejpam-5974	264	12	of	of	ADP
ejpam-5974	264	13	bn	bn	NOUN
ejpam-5974	264	14	,	,	PUNCT
ejpam-5974	264	15	k	k	PROPN
ejpam-5974	264	16	are	be	AUX
ejpam-5974	264	17	nk	nk	PROPN
ejpam-5974	264	18	and	and	CCONJ
ejpam-5974	264	19	nk	nk	PROPN
ejpam-5974	265	1	+	+	CCONJ
ejpam-5974	265	2	1	1	NUM
ejpam-5974	265	3	respectively	respectively	ADV
ejpam-5974	265	4	,	,	PUNCT
ejpam-5974	265	5	we	we	PRON
ejpam-5974	265	6	have	have	VERB
ejpam-5974	265	7	d(µ(bn	d(µ(bn	NOUN
ejpam-5974	265	8	,	,	PUNCT
ejpam-5974	265	9	k	k	NOUN
ejpam-5974	265	10	)	)	PUNCT
ejpam-5974	265	11	)	)	PUNCT
ejpam-5974	266	1	=	=	SYM
ejpam-5974	266	2	3|e(bn	3|e(bn	PROPN
ejpam-5974	266	3	,	,	PUNCT
ejpam-5974	266	4	k)|+	k)|+	ADJ
ejpam-5974	266	5	|v	|v	X
ejpam-5974	266	6	(	(	PUNCT
ejpam-5974	266	7	bn	bn	INTJ
ejpam-5974	266	8	,	,	PUNCT
ejpam-5974	266	9	k)|	k)|	PROPN
ejpam-5974	266	10	|v	|v	PROPN
ejpam-5974	266	11	(	(	PUNCT
ejpam-5974	266	12	bn	bn	X
ejpam-5974	266	13	,	,	PUNCT
ejpam-5974	266	14	k)|(2|v	k)|(2|v	PROPN
ejpam-5974	266	15	(	(	PUNCT
ejpam-5974	266	16	bn	bn	INTJ
ejpam-5974	266	17	,	,	PUNCT
ejpam-5974	266	18	k)|+	k)|+	NOUN
ejpam-5974	266	19	1	1	NUM
ejpam-5974	266	20	)	)	PUNCT
ejpam-5974	266	21	=	=	SYM
ejpam-5974	266	22	3nk	3nk	NOUN
ejpam-5974	267	1	+	+	CCONJ
ejpam-5974	267	2	nk	nk	PROPN
ejpam-5974	267	3	+	+	NOUN
ejpam-5974	267	4	1	1	NUM
ejpam-5974	267	5	(	(	PUNCT
ejpam-5974	267	6	nk	nk	PROPN
ejpam-5974	267	7	+	+	NOUN
ejpam-5974	267	8	1)[2(nk	1)[2(nk	NUM
ejpam-5974	267	9	+	+	CCONJ
ejpam-5974	267	10	1	1	NUM
ejpam-5974	267	11	)	)	PUNCT
ejpam-5974	267	12	+	+	CCONJ
ejpam-5974	267	13	1	1	X
ejpam-5974	267	14	]	]	PUNCT
ejpam-5974	267	15	=	=	PUNCT
ejpam-5974	267	16	4nk	4nk	NOUN
ejpam-5974	268	1	+	+	CCONJ
ejpam-5974	268	2	1	1	NUM
ejpam-5974	268	3	(	(	PUNCT
ejpam-5974	268	4	nk	nk	PROPN
ejpam-5974	268	5	+	+	PROPN
ejpam-5974	268	6	1)(2nk	1)(2nk	NUM
ejpam-5974	268	7	+	+	CCONJ
ejpam-5974	268	8	3	3	NUM
ejpam-5974	268	9	)	)	PUNCT
ejpam-5974	268	10	=	=	PUNCT
ejpam-5974	268	11	4nk	4nk	NOUN
ejpam-5974	269	1	+	+	CCONJ
ejpam-5974	270	1	1	1	NUM
ejpam-5974	270	2	2n2k2	2n2k2	NUM
ejpam-5974	270	3	+	+	CCONJ
ejpam-5974	270	4	5nk	5nk	ADJ
ejpam-5974	270	5	+	+	CCONJ
ejpam-5974	270	6	3	3	NUM
ejpam-5974	270	7	.	.	PUNCT
ejpam-5974	271	1	corollary	corollary	ADJ
ejpam-5974	271	2	8	8	NUM
ejpam-5974	271	3	.	.	PUNCT
ejpam-5974	272	1	let	let	VERB
ejpam-5974	272	2	g	g	PRON
ejpam-5974	272	3	be	be	AUX
ejpam-5974	272	4	a	a	DET
ejpam-5974	272	5	lollipop	lollipop	NOUN
ejpam-5974	272	6	graph	graph	NOUN
ejpam-5974	272	7	(	(	PUNCT
ejpam-5974	272	8	lm	lm	INTJ
ejpam-5974	272	9	,	,	PUNCT
ejpam-5974	272	10	n	n	CCONJ
ejpam-5974	272	11	)	)	PUNCT
ejpam-5974	273	1	where	where	SCONJ
ejpam-5974	273	2	m	m	PROPN
ejpam-5974	273	3	≥	≥	VERB
ejpam-5974	273	4	3	3	NUM
ejpam-5974	273	5	and	and	CCONJ
ejpam-5974	273	6	n	n	PRON
ejpam-5974	273	7	≥	≥	NOUN
ejpam-5974	273	8	1	1	NUM
ejpam-5974	273	9	.	.	PUNCT
ejpam-5974	273	10	then	then	ADV
ejpam-5974	273	11	d(µ(lm	d(µ(lm	PROPN
ejpam-5974	273	12	,	,	PUNCT
ejpam-5974	273	13	n	n	CCONJ
ejpam-5974	273	14	)	)	PUNCT
ejpam-5974	273	15	)	)	PUNCT
ejpam-5974	274	1	=	=	PUNCT
ejpam-5974	275	1	3m2	3m2	NUM
ejpam-5974	275	2	−m+	−m+	NOUN
ejpam-5974	275	3	8n	8n	NOUN
ejpam-5974	275	4	4m2	4m2	NUM
ejpam-5974	275	5	+	+	CCONJ
ejpam-5974	275	6	8mn+	8mn+	NOUN
ejpam-5974	275	7	4n2	4n2	NUM
ejpam-5974	276	1	+	+	NOUN
ejpam-5974	276	2	m+	m+	NOUN
ejpam-5974	276	3	n	n	NOUN
ejpam-5974	276	4	.	.	PUNCT
ejpam-5974	277	1	proof	proof	NOUN
ejpam-5974	277	2	.	.	PUNCT
ejpam-5974	278	1	by	by	ADP
ejpam-5974	278	2	theorem	theorem	NOUN
ejpam-5974	278	3	10	10	NUM
ejpam-5974	278	4	and	and	CCONJ
ejpam-5974	278	5	since	since	SCONJ
ejpam-5974	278	6	the	the	DET
ejpam-5974	278	7	number	number	NOUN
ejpam-5974	278	8	of	of	ADP
ejpam-5974	278	9	edges	edge	NOUN
ejpam-5974	278	10	and	and	CCONJ
ejpam-5974	278	11	vertices	vertex	NOUN
ejpam-5974	278	12	of	of	ADP
ejpam-5974	278	13	lm	lm	PROPN
ejpam-5974	278	14	,	,	PUNCT
ejpam-5974	278	15	n	n	PRON
ejpam-5974	278	16	are	be	AUX
ejpam-5974	278	17	m(m−	m(m−	PROPN
ejpam-5974	278	18	1	1	NUM
ejpam-5974	278	19	)	)	PUNCT
ejpam-5974	278	20	+	+	CCONJ
ejpam-5974	278	21	2	2	NUM
ejpam-5974	278	22	2	2	NUM
ejpam-5974	278	23	and	and	CCONJ
ejpam-5974	278	24	m+	m+	NUM
ejpam-5974	278	25	n	n	CCONJ
ejpam-5974	278	26	respectively	respectively	ADV
ejpam-5974	278	27	,	,	PUNCT
ejpam-5974	278	28	we	we	PRON
ejpam-5974	278	29	have	have	VERB
ejpam-5974	278	30	d(µ(lm	d(µ(lm	NOUN
ejpam-5974	278	31	,	,	PUNCT
ejpam-5974	278	32	n	n	CCONJ
ejpam-5974	278	33	)	)	PUNCT
ejpam-5974	278	34	)	)	PUNCT
ejpam-5974	279	1	=	=	SYM
ejpam-5974	279	2	3|e(lm	3|e(lm	ADV
ejpam-5974	279	3	,	,	PUNCT
ejpam-5974	279	4	n)|+	n)|+	PROPN
ejpam-5974	279	5	|v	|v	PROPN
ejpam-5974	279	6	(	(	PUNCT
ejpam-5974	279	7	lm	lm	INTJ
ejpam-5974	279	8	,	,	PUNCT
ejpam-5974	279	9	n)|	n)|	PROPN
ejpam-5974	279	10	|v	|v	PROPN
ejpam-5974	279	11	(	(	PUNCT
ejpam-5974	279	12	lm	lm	INTJ
ejpam-5974	279	13	,	,	PUNCT
ejpam-5974	279	14	n)|(2|v	n)|(2|v	PROPN
ejpam-5974	279	15	(	(	PUNCT
ejpam-5974	279	16	lm	lm	INTJ
ejpam-5974	279	17	,	,	PUNCT
ejpam-5974	279	18	n)|+	n)|+	PROPN
ejpam-5974	279	19	1	1	NUM
ejpam-5974	279	20	)	)	PUNCT
ejpam-5974	279	21	=	=	SYM
ejpam-5974	279	22	3	3	NUM
ejpam-5974	279	23	(	(	PUNCT
ejpam-5974	279	24	m(m−	m(m−	PROPN
ejpam-5974	279	25	1	1	NUM
ejpam-5974	279	26	)	)	PUNCT
ejpam-5974	279	27	+	+	NUM
ejpam-5974	279	28	2n	2n	NUM
ejpam-5974	279	29	2	2	NUM
ejpam-5974	279	30	)	)	PUNCT
ejpam-5974	280	1	+	+	NOUN
ejpam-5974	280	2	m+	m+	NUM
ejpam-5974	280	3	n	n	CCONJ
ejpam-5974	280	4	(	(	PUNCT
ejpam-5974	280	5	m+	m+	NUM
ejpam-5974	280	6	n)[2(m+	n)[2(m+	ADJ
ejpam-5974	280	7	n	n	CCONJ
ejpam-5974	280	8	)	)	PUNCT
ejpam-5974	280	9	+	+	CCONJ
ejpam-5974	280	10	1	1	X
ejpam-5974	280	11	]	]	X
ejpam-5974	280	12	=	=	SYM
ejpam-5974	280	13	3(m2	3(m2	NUM
ejpam-5974	280	14	−m+	−m+	NOUN
ejpam-5974	280	15	2n	2n	NUM
ejpam-5974	280	16	)	)	PUNCT
ejpam-5974	280	17	2	2	NUM
ejpam-5974	281	1	+	+	NUM
ejpam-5974	281	2	m+	m+	NUM
ejpam-5974	281	3	n	n	CCONJ
ejpam-5974	281	4	(	(	PUNCT
ejpam-5974	281	5	m+	m+	NOUN
ejpam-5974	281	6	n)(2m+	n)(2m+	PROPN
ejpam-5974	281	7	2n+	2n+	NUM
ejpam-5974	281	8	1	1	NUM
ejpam-5974	281	9	)	)	PUNCT
ejpam-5974	281	10	=	=	PUNCT
ejpam-5974	282	1	3m2	3m2	NUM
ejpam-5974	282	2	−	−	NOUN
ejpam-5974	282	3	3m+	3m+	NUM
ejpam-5974	282	4	6n+	6n+	NUM
ejpam-5974	282	5	2m+	2m+	NUM
ejpam-5974	282	6	2n	2n	NUM
ejpam-5974	282	7	2	2	NUM
ejpam-5974	282	8	2m2	2m2	NUM
ejpam-5974	282	9	+	+	CCONJ
ejpam-5974	282	10	4mn+	4mn+	NOUN
ejpam-5974	282	11	2n2	2n2	NUM
ejpam-5974	283	1	+	+	SYM
ejpam-5974	283	2	m+	m+	NUM
ejpam-5974	283	3	n	n	NOUN
ejpam-5974	283	4	=	=	SYM
ejpam-5974	283	5	3m2	3m2	NOUN
ejpam-5974	284	1	−m+	−m+	NOUN
ejpam-5974	284	2	8n	8n	NOUN
ejpam-5974	284	3	2	2	NUM
ejpam-5974	284	4	2m2	2m2	NUM
ejpam-5974	284	5	+	+	CCONJ
ejpam-5974	284	6	4mn+	4mn+	NOUN
ejpam-5974	284	7	2n2	2n2	NUM
ejpam-5974	285	1	+	+	SYM
ejpam-5974	285	2	m+	m+	NUM
ejpam-5974	285	3	n	n	NOUN
ejpam-5974	285	4	=	=	SYM
ejpam-5974	285	5	3m2	3m2	NUM
ejpam-5974	285	6	−m+	−m+	NOUN
ejpam-5974	285	7	8n	8n	NOUN
ejpam-5974	285	8	4m2	4m2	NUM
ejpam-5974	286	1	+	+	CCONJ
ejpam-5974	286	2	8mn+	8mn+	NOUN
ejpam-5974	286	3	4n2	4n2	NUM
ejpam-5974	287	1	+	+	NOUN
ejpam-5974	287	2	m+	m+	NOUN
ejpam-5974	287	3	n	n	PROPN
ejpam-5974	287	4	.	.	PUNCT
ejpam-5974	288	1	r.	r.	PROPN
ejpam-5974	288	2	sango	sango	PROPN
ejpam-5974	288	3	,	,	PUNCT
ejpam-5974	288	4	i.	i.	NOUN
ejpam-5974	288	5	cabahug	cabahug	PROPN
ejpam-5974	288	6	,	,	PUNCT
ejpam-5974	288	7	jr	jr	PROPN
ejpam-5974	288	8	/	/	SYM
ejpam-5974	288	9	eur	eur	PROPN
ejpam-5974	288	10	.	.	PUNCT
ejpam-5974	289	1	j.	j.	PROPN
ejpam-5974	289	2	pure	pure	PROPN
ejpam-5974	289	3	appl	appl	PROPN
ejpam-5974	289	4	.	.	PROPN
ejpam-5974	289	5	math	math	PROPN
ejpam-5974	289	6	,	,	PUNCT
ejpam-5974	289	7	18	18	NUM
ejpam-5974	289	8	(	(	PUNCT
ejpam-5974	289	9	3	3	NUM
ejpam-5974	289	10	)	)	PUNCT
ejpam-5974	289	11	(	(	PUNCT
ejpam-5974	289	12	2025	2025	NUM
ejpam-5974	289	13	)	)	PUNCT
ejpam-5974	289	14	,	,	PUNCT
ejpam-5974	289	15	5974	5974	NUM
ejpam-5974	289	16	14	14	NUM
ejpam-5974	289	17	of	of	ADP
ejpam-5974	289	18	16	16	NUM
ejpam-5974	289	19	corollary	corollary	ADJ
ejpam-5974	289	20	9	9	NUM
ejpam-5974	289	21	.	.	PUNCT
ejpam-5974	290	1	let	let	VERB
ejpam-5974	290	2	g	g	PRON
ejpam-5974	290	3	be	be	AUX
ejpam-5974	290	4	a	a	DET
ejpam-5974	290	5	gear	gear	NOUN
ejpam-5974	290	6	graph	graph	NOUN
ejpam-5974	290	7	(	(	PUNCT
ejpam-5974	290	8	gn	gn	PROPN
ejpam-5974	290	9	)	)	PUNCT
ejpam-5974	290	10	where	where	SCONJ
ejpam-5974	290	11	n	n	PRON
ejpam-5974	290	12	≥	≥	NOUN
ejpam-5974	290	13	3	3	NUM
ejpam-5974	290	14	.	.	PUNCT
ejpam-5974	290	15	then	then	ADV
ejpam-5974	290	16	d(µ(gn	d(µ(gn	NOUN
ejpam-5974	290	17	)	)	PUNCT
ejpam-5974	290	18	)	)	PUNCT
ejpam-5974	291	1	=	=	PUNCT
ejpam-5974	291	2	11n+	11n+	NUM
ejpam-5974	291	3	1	1	NUM
ejpam-5974	291	4	8n2	8n2	NUM
ejpam-5974	291	5	+	+	CCONJ
ejpam-5974	291	6	10n+	10n+	NUM
ejpam-5974	291	7	3	3	NUM
ejpam-5974	291	8	.	.	PUNCT
ejpam-5974	292	1	proof	proof	NOUN
ejpam-5974	292	2	.	.	PUNCT
ejpam-5974	293	1	by	by	ADP
ejpam-5974	293	2	theorem	theorem	NOUN
ejpam-5974	293	3	10	10	NUM
ejpam-5974	293	4	and	and	CCONJ
ejpam-5974	293	5	since	since	SCONJ
ejpam-5974	293	6	the	the	DET
ejpam-5974	293	7	number	number	NOUN
ejpam-5974	293	8	of	of	ADP
ejpam-5974	293	9	edges	edge	NOUN
ejpam-5974	293	10	and	and	CCONJ
ejpam-5974	293	11	vertices	vertex	NOUN
ejpam-5974	293	12	of	of	ADP
ejpam-5974	293	13	gn	gn	PROPN
ejpam-5974	293	14	are	be	AUX
ejpam-5974	293	15	3n	3n	NUM
ejpam-5974	293	16	and	and	CCONJ
ejpam-5974	293	17	2n+	2n+	NUM
ejpam-5974	293	18	1	1	NUM
ejpam-5974	293	19	respectively	respectively	ADV
ejpam-5974	293	20	,	,	PUNCT
ejpam-5974	293	21	we	we	PRON
ejpam-5974	293	22	have	have	VERB
ejpam-5974	293	23	d(µ(gn	d(µ(gn	NOUN
ejpam-5974	293	24	)	)	PUNCT
ejpam-5974	293	25	)	)	PUNCT
ejpam-5974	294	1	=	=	SYM
ejpam-5974	295	1	3|e(gn)|+	3|e(gn)|+	NUM
ejpam-5974	295	2	|v	|v	NOUN
ejpam-5974	295	3	(	(	PUNCT
ejpam-5974	295	4	gn)|	gn)|	PROPN
ejpam-5974	295	5	|v	|v	PROPN
ejpam-5974	295	6	(	(	PUNCT
ejpam-5974	295	7	gn)|(2|v	gn)|(2|v	PROPN
ejpam-5974	295	8	(	(	PUNCT
ejpam-5974	295	9	gn)|+	gn)|+	PROPN
ejpam-5974	295	10	1	1	NUM
ejpam-5974	295	11	)	)	PUNCT
ejpam-5974	295	12	=	=	SYM
ejpam-5974	295	13	3(3n	3(3n	NUM
ejpam-5974	295	14	)	)	PUNCT
ejpam-5974	295	15	+	+	CCONJ
ejpam-5974	295	16	2n+	2n+	NUM
ejpam-5974	295	17	1	1	NUM
ejpam-5974	295	18	(	(	PUNCT
ejpam-5974	295	19	2n+	2n+	NUM
ejpam-5974	295	20	1)[2(2n+	1)[2(2n+	NUM
ejpam-5974	295	21	1	1	NUM
ejpam-5974	295	22	)	)	PUNCT
ejpam-5974	295	23	+	+	CCONJ
ejpam-5974	295	24	1	1	X
ejpam-5974	295	25	]	]	X
ejpam-5974	295	26	=	=	SYM
ejpam-5974	295	27	9n+	9n+	NUM
ejpam-5974	295	28	2n+	2n+	NUM
ejpam-5974	295	29	1	1	NUM
ejpam-5974	295	30	(	(	PUNCT
ejpam-5974	295	31	2n+	2n+	NUM
ejpam-5974	295	32	1)(4n+	1)(4n+	NOUN
ejpam-5974	295	33	3	3	NUM
ejpam-5974	295	34	)	)	PUNCT
ejpam-5974	295	35	=	=	PUNCT
ejpam-5974	295	36	11n+	11n+	NUM
ejpam-5974	295	37	1	1	NUM
ejpam-5974	295	38	8n2	8n2	NUM
ejpam-5974	295	39	+	+	CCONJ
ejpam-5974	295	40	6n+	6n+	NUM
ejpam-5974	295	41	4n+	4n+	NUM
ejpam-5974	295	42	3	3	NUM
ejpam-5974	295	43	=	=	SYM
ejpam-5974	295	44	11n+	11n+	NUM
ejpam-5974	295	45	1	1	NUM
ejpam-5974	295	46	8n2	8n2	NUM
ejpam-5974	295	47	+	+	CCONJ
ejpam-5974	295	48	10n+	10n+	NUM
ejpam-5974	295	49	3	3	NUM
ejpam-5974	295	50	.	.	PUNCT
ejpam-5974	296	1	corollary	corollary	ADJ
ejpam-5974	296	2	10	10	NUM
ejpam-5974	296	3	.	.	PUNCT
ejpam-5974	297	1	let	let	VERB
ejpam-5974	297	2	g	g	PRON
ejpam-5974	297	3	be	be	AUX
ejpam-5974	297	4	a	a	DET
ejpam-5974	297	5	tadpole	tadpole	NOUN
ejpam-5974	297	6	graph	graph	NOUN
ejpam-5974	297	7	(	(	PUNCT
ejpam-5974	297	8	tm	tm	NOUN
ejpam-5974	297	9	,	,	PUNCT
ejpam-5974	297	10	n	n	CCONJ
ejpam-5974	297	11	)	)	PUNCT
ejpam-5974	298	1	where	where	SCONJ
ejpam-5974	298	2	m	m	PROPN
ejpam-5974	298	3	≥	≥	VERB
ejpam-5974	298	4	3	3	NUM
ejpam-5974	298	5	and	and	CCONJ
ejpam-5974	298	6	n	n	PRON
ejpam-5974	298	7	≥	≥	NOUN
ejpam-5974	298	8	1	1	NUM
ejpam-5974	298	9	.	.	PUNCT
ejpam-5974	298	10	then	then	ADV
ejpam-5974	298	11	,	,	PUNCT
ejpam-5974	298	12	d(µ(tm	d(µ(tm	NOUN
ejpam-5974	298	13	,	,	PUNCT
ejpam-5974	298	14	n	n	CCONJ
ejpam-5974	298	15	)	)	PUNCT
ejpam-5974	298	16	)	)	PUNCT
ejpam-5974	299	1	=	=	SYM
ejpam-5974	299	2	4	4	NUM
ejpam-5974	299	3	2m+	2m+	NUM
ejpam-5974	299	4	2n+	2n+	NUM
ejpam-5974	299	5	1	1	NUM
ejpam-5974	299	6	.	.	PUNCT
ejpam-5974	300	1	proof	proof	NOUN
ejpam-5974	300	2	.	.	PUNCT
ejpam-5974	301	1	by	by	ADP
ejpam-5974	301	2	theorem	theorem	NOUN
ejpam-5974	301	3	10	10	NUM
ejpam-5974	301	4	and	and	CCONJ
ejpam-5974	301	5	since	since	SCONJ
ejpam-5974	301	6	the	the	DET
ejpam-5974	301	7	number	number	NOUN
ejpam-5974	301	8	of	of	ADP
ejpam-5974	301	9	edges	edge	NOUN
ejpam-5974	301	10	and	and	CCONJ
ejpam-5974	301	11	vertices	vertex	NOUN
ejpam-5974	301	12	of	of	ADP
ejpam-5974	301	13	tm	tm	PROPN
ejpam-5974	301	14	,	,	PUNCT
ejpam-5974	301	15	n	n	PROPN
ejpam-5974	301	16	is	be	AUX
ejpam-5974	301	17	m	m	PRON
ejpam-5974	301	18	+	+	NOUN
ejpam-5974	301	19	n	n	CCONJ
ejpam-5974	301	20	,	,	PUNCT
ejpam-5974	301	21	we	we	PRON
ejpam-5974	301	22	have	have	VERB
ejpam-5974	301	23	d(µ(tm	d(µ(tm	NOUN
ejpam-5974	301	24	,	,	PUNCT
ejpam-5974	301	25	n	n	CCONJ
ejpam-5974	301	26	)	)	PUNCT
ejpam-5974	301	27	)	)	PUNCT
ejpam-5974	302	1	=	=	SYM
ejpam-5974	302	2	3|e(tm	3|e(tm	NOUN
ejpam-5974	302	3	,	,	PUNCT
ejpam-5974	302	4	n)|+	n)|+	PROPN
ejpam-5974	302	5	|v	|v	PROPN
ejpam-5974	302	6	(	(	PUNCT
ejpam-5974	302	7	tm	tm	PROPN
ejpam-5974	302	8	,	,	PUNCT
ejpam-5974	302	9	n)|	n)|	PROPN
ejpam-5974	302	10	|v	|v	PROPN
ejpam-5974	302	11	(	(	PUNCT
ejpam-5974	302	12	tm	tm	NOUN
ejpam-5974	302	13	,	,	PUNCT
ejpam-5974	302	14	n)|(2|v	n)|(2|v	PROPN
ejpam-5974	302	15	(	(	PUNCT
ejpam-5974	302	16	tm	tm	PROPN
ejpam-5974	302	17	,	,	PUNCT
ejpam-5974	302	18	n)|+	n)|+	PROPN
ejpam-5974	302	19	1	1	NUM
ejpam-5974	302	20	)	)	PUNCT
ejpam-5974	302	21	=	=	SYM
ejpam-5974	302	22	3(m+	3(m+	NUM
ejpam-5974	302	23	n	n	CCONJ
ejpam-5974	302	24	)	)	PUNCT
ejpam-5974	303	1	+	+	NOUN
ejpam-5974	303	2	m+	m+	NUM
ejpam-5974	303	3	n	n	CCONJ
ejpam-5974	303	4	(	(	PUNCT
ejpam-5974	303	5	m+	m+	NUM
ejpam-5974	303	6	n)[2(m+	n)[2(m+	ADJ
ejpam-5974	303	7	n	n	CCONJ
ejpam-5974	303	8	)	)	PUNCT
ejpam-5974	303	9	+	+	CCONJ
ejpam-5974	303	10	1	1	X
ejpam-5974	303	11	]	]	PUNCT
ejpam-5974	303	12	=	=	SYM
ejpam-5974	303	13	4m+	4m+	NUM
ejpam-5974	303	14	4n	4n	X
ejpam-5974	303	15	(	(	PUNCT
ejpam-5974	303	16	m+	m+	NOUN
ejpam-5974	303	17	n)(2m+	n)(2m+	PROPN
ejpam-5974	303	18	2n+	2n+	NUM
ejpam-5974	303	19	1	1	NUM
ejpam-5974	303	20	)	)	PUNCT
ejpam-5974	303	21	=	=	SYM
ejpam-5974	303	22	4(m+	4(m+	NUM
ejpam-5974	303	23	n	n	CCONJ
ejpam-5974	303	24	)	)	PUNCT
ejpam-5974	303	25	(	(	PUNCT
ejpam-5974	303	26	m+	m+	NOUN
ejpam-5974	303	27	n)(2m+	n)(2m+	PROPN
ejpam-5974	303	28	2n+	2n+	NUM
ejpam-5974	303	29	1	1	NUM
ejpam-5974	303	30	)	)	PUNCT
ejpam-5974	303	31	=	=	SYM
ejpam-5974	303	32	4	4	NUM
ejpam-5974	303	33	2m+	2m+	NUM
ejpam-5974	303	34	2n+	2n+	NUM
ejpam-5974	303	35	1	1	NUM
ejpam-5974	303	36	.	.	PUNCT
ejpam-5974	304	1	corollary	corollary	ADJ
ejpam-5974	304	2	11	11	NUM
ejpam-5974	304	3	.	.	PUNCT
ejpam-5974	305	1	let	let	VERB
ejpam-5974	305	2	g	g	PRON
ejpam-5974	305	3	be	be	AUX
ejpam-5974	305	4	a	a	DET
ejpam-5974	305	5	wheel	wheel	NOUN
ejpam-5974	305	6	graph	graph	NOUN
ejpam-5974	305	7	(	(	PUNCT
ejpam-5974	305	8	wn	wn	PROPN
ejpam-5974	305	9	)	)	PUNCT
ejpam-5974	305	10	where	where	SCONJ
ejpam-5974	305	11	n	n	PRON
ejpam-5974	305	12	≥	≥	NOUN
ejpam-5974	305	13	3	3	NUM
ejpam-5974	305	14	.	.	PUNCT
ejpam-5974	306	1	then	then	ADV
ejpam-5974	306	2	d(µ(wn	d(µ(wn	NOUN
ejpam-5974	306	3	)	)	PUNCT
ejpam-5974	306	4	)	)	PUNCT
ejpam-5974	307	1	=	=	PUNCT
ejpam-5974	308	1	7n−	7n−	NUM
ejpam-5974	308	2	6	6	NUM
ejpam-5974	308	3	2n2	2n2	NUM
ejpam-5974	308	4	+	+	CCONJ
ejpam-5974	308	5	n	n	NOUN
ejpam-5974	308	6	.	.	PUNCT
ejpam-5974	309	1	proof	proof	NOUN
ejpam-5974	309	2	.	.	PUNCT
ejpam-5974	310	1	by	by	ADP
ejpam-5974	310	2	theorem	theorem	NOUN
ejpam-5974	310	3	10	10	NUM
ejpam-5974	310	4	and	and	CCONJ
ejpam-5974	310	5	since	since	SCONJ
ejpam-5974	310	6	the	the	DET
ejpam-5974	310	7	number	number	NOUN
ejpam-5974	310	8	of	of	ADP
ejpam-5974	310	9	edges	edge	NOUN
ejpam-5974	310	10	and	and	CCONJ
ejpam-5974	310	11	vertices	vertex	NOUN
ejpam-5974	310	12	of	of	ADP
ejpam-5974	310	13	wn	wn	PROPN
ejpam-5974	310	14	are	be	AUX
ejpam-5974	310	15	2n	2n	NUM
ejpam-5974	310	16	−	−	ADP
ejpam-5974	310	17	2	2	NUM
ejpam-5974	310	18	and	and	CCONJ
ejpam-5974	310	19	n	n	PRON
ejpam-5974	310	20	respectively	respectively	ADV
ejpam-5974	310	21	,	,	PUNCT
ejpam-5974	310	22	we	we	PRON
ejpam-5974	310	23	have	have	VERB
ejpam-5974	310	24	d(µ(wn	d(µ(wn	NOUN
ejpam-5974	310	25	)	)	PUNCT
ejpam-5974	310	26	)	)	PUNCT
ejpam-5974	311	1	=	=	PUNCT
ejpam-5974	311	2	3|e(wn)|+	3|e(wn)|+	NUM
ejpam-5974	311	3	|v	|v	NOUN
ejpam-5974	311	4	(	(	PUNCT
ejpam-5974	311	5	wn)|	wn)|	PROPN
ejpam-5974	311	6	|v	|v	PROPN
ejpam-5974	311	7	(	(	PUNCT
ejpam-5974	311	8	wn)|(2|v	wn)|(2|v	PROPN
ejpam-5974	311	9	(	(	PUNCT
ejpam-5974	311	10	wn)|+	wn)|+	NUM
ejpam-5974	311	11	1	1	NUM
ejpam-5974	311	12	)	)	PUNCT
ejpam-5974	311	13	=	=	SYM
ejpam-5974	311	14	3(2n−	3(2n−	NUM
ejpam-5974	311	15	2	2	NUM
ejpam-5974	311	16	)	)	PUNCT
ejpam-5974	311	17	+	+	CCONJ
ejpam-5974	311	18	n	n	CCONJ
ejpam-5974	311	19	n(2n+	n(2n+	NOUN
ejpam-5974	311	20	1	1	NUM
ejpam-5974	311	21	)	)	PUNCT
ejpam-5974	311	22	=	=	SYM
ejpam-5974	312	1	6n−	6n−	NUM
ejpam-5974	312	2	6	6	NUM
ejpam-5974	312	3	+	+	NOUN
ejpam-5974	312	4	n	n	PRON
ejpam-5974	312	5	2n2	2n2	NUM
ejpam-5974	312	6	+	+	CCONJ
ejpam-5974	312	7	n	n	NOUN
ejpam-5974	312	8	=	=	SYM
ejpam-5974	312	9	7n−	7n−	NUM
ejpam-5974	312	10	6	6	NUM
ejpam-5974	312	11	2n2	2n2	NUM
ejpam-5974	312	12	+	+	CCONJ
ejpam-5974	312	13	n	n	NOUN
ejpam-5974	312	14	.	.	PUNCT
ejpam-5974	313	1	corollary	corollary	ADJ
ejpam-5974	313	2	12	12	NUM
ejpam-5974	313	3	.	.	PUNCT
ejpam-5974	314	1	let	let	VERB
ejpam-5974	314	2	g	g	PRON
ejpam-5974	314	3	be	be	AUX
ejpam-5974	314	4	a	a	DET
ejpam-5974	314	5	fan	fan	NOUN
ejpam-5974	314	6	graph	graph	NOUN
ejpam-5974	314	7	(	(	PUNCT
ejpam-5974	314	8	fn	fn	NOUN
ejpam-5974	314	9	)	)	PUNCT
ejpam-5974	314	10	where	where	SCONJ
ejpam-5974	314	11	n	n	PRON
ejpam-5974	314	12	≥	≥	NOUN
ejpam-5974	314	13	3	3	NUM
ejpam-5974	314	14	.	.	PUNCT
ejpam-5974	315	1	then	then	ADV
ejpam-5974	315	2	d(µ(fn	d(µ(fn	NOUN
ejpam-5974	315	3	)	)	PUNCT
ejpam-5974	315	4	)	)	PUNCT
ejpam-5974	316	1	=	=	PUNCT
ejpam-5974	317	1	7n−	7n−	NUM
ejpam-5974	317	2	2	2	NUM
ejpam-5974	317	3	2n2	2n2	NUM
ejpam-5974	317	4	+	+	CCONJ
ejpam-5974	317	5	5n+	5n+	NUM
ejpam-5974	317	6	3	3	NUM
ejpam-5974	317	7	.	.	PUNCT
ejpam-5974	318	1	r.	r.	PROPN
ejpam-5974	318	2	sango	sango	PROPN
ejpam-5974	318	3	,	,	PUNCT
ejpam-5974	318	4	i.	i.	NOUN
ejpam-5974	318	5	cabahug	cabahug	PROPN
ejpam-5974	318	6	,	,	PUNCT
ejpam-5974	318	7	jr	jr	PROPN
ejpam-5974	318	8	/	/	SYM
ejpam-5974	318	9	eur	eur	PROPN
ejpam-5974	318	10	.	.	PUNCT
ejpam-5974	319	1	j.	j.	PROPN
ejpam-5974	319	2	pure	pure	PROPN
ejpam-5974	319	3	appl	appl	PROPN
ejpam-5974	319	4	.	.	PROPN
ejpam-5974	319	5	math	math	PROPN
ejpam-5974	319	6	,	,	PUNCT
ejpam-5974	319	7	18	18	NUM
ejpam-5974	319	8	(	(	PUNCT
ejpam-5974	319	9	3	3	NUM
ejpam-5974	319	10	)	)	PUNCT
ejpam-5974	319	11	(	(	PUNCT
ejpam-5974	319	12	2025	2025	NUM
ejpam-5974	319	13	)	)	PUNCT
ejpam-5974	319	14	,	,	PUNCT
ejpam-5974	319	15	5974	5974	NUM
ejpam-5974	319	16	15	15	NUM
ejpam-5974	319	17	of	of	ADP
ejpam-5974	319	18	16	16	NUM
ejpam-5974	319	19	proof	proof	NOUN
ejpam-5974	319	20	.	.	PUNCT
ejpam-5974	320	1	by	by	ADP
ejpam-5974	320	2	theorem	theorem	NOUN
ejpam-5974	320	3	10	10	NUM
ejpam-5974	320	4	and	and	CCONJ
ejpam-5974	320	5	since	since	SCONJ
ejpam-5974	320	6	the	the	DET
ejpam-5974	320	7	number	number	NOUN
ejpam-5974	320	8	of	of	ADP
ejpam-5974	320	9	edges	edge	NOUN
ejpam-5974	320	10	and	and	CCONJ
ejpam-5974	320	11	vertices	vertex	NOUN
ejpam-5974	320	12	of	of	ADP
ejpam-5974	320	13	fn	fn	NOUN
ejpam-5974	320	14	are	be	AUX
ejpam-5974	320	15	2n	2n	NUM
ejpam-5974	320	16	−	−	ADP
ejpam-5974	320	17	1	1	NUM
ejpam-5974	320	18	and	and	CCONJ
ejpam-5974	320	19	n+	n+	NUM
ejpam-5974	320	20	1	1	NUM
ejpam-5974	320	21	respectively	respectively	ADV
ejpam-5974	320	22	,	,	PUNCT
ejpam-5974	320	23	we	we	PRON
ejpam-5974	320	24	have	have	VERB
ejpam-5974	320	25	d(µ(fn	d(µ(fn	NOUN
ejpam-5974	320	26	)	)	PUNCT
ejpam-5974	320	27	)	)	PUNCT
ejpam-5974	321	1	=	=	SYM
ejpam-5974	322	1	3|e(fn)|+	3|e(fn)|+	NUM
ejpam-5974	322	2	|v	|v	NOUN
ejpam-5974	322	3	(	(	PUNCT
ejpam-5974	322	4	fn)|	fn)|	PROPN
ejpam-5974	322	5	|v	|v	PROPN
ejpam-5974	322	6	(	(	PUNCT
ejpam-5974	322	7	fn)|(2|v	fn)|(2|v	PROPN
ejpam-5974	322	8	(	(	PUNCT
ejpam-5974	322	9	fn)|+	fn)|+	NOUN
ejpam-5974	322	10	1	1	NUM
ejpam-5974	322	11	)	)	PUNCT
ejpam-5974	322	12	=	=	SYM
ejpam-5974	322	13	3(2n−	3(2n−	NUM
ejpam-5974	322	14	1	1	NUM
ejpam-5974	322	15	)	)	PUNCT
ejpam-5974	322	16	+	+	CCONJ
ejpam-5974	322	17	n+	n+	NUM
ejpam-5974	322	18	1	1	NUM
ejpam-5974	322	19	(	(	PUNCT
ejpam-5974	322	20	n+	n+	NUM
ejpam-5974	322	21	1)[2(n+	1)[2(n+	NUM
ejpam-5974	322	22	1	1	NUM
ejpam-5974	322	23	)	)	PUNCT
ejpam-5974	322	24	+	+	CCONJ
ejpam-5974	322	25	1	1	X
ejpam-5974	322	26	]	]	X
ejpam-5974	322	27	=	=	SYM
ejpam-5974	322	28	6n−	6n−	NUM
ejpam-5974	322	29	3	3	NUM
ejpam-5974	322	30	+	+	CCONJ
ejpam-5974	322	31	n+	n+	NUM
ejpam-5974	322	32	1	1	NUM
ejpam-5974	322	33	(	(	PUNCT
ejpam-5974	322	34	n+	n+	NUM
ejpam-5974	322	35	1)(2n+	1)(2n+	NUM
ejpam-5974	322	36	3	3	X
ejpam-5974	322	37	)	)	PUNCT
ejpam-5974	322	38	=	=	SYM
ejpam-5974	323	1	7n−	7n−	NUM
ejpam-5974	323	2	2	2	NUM
ejpam-5974	323	3	2n2	2n2	NUM
ejpam-5974	323	4	+	+	CCONJ
ejpam-5974	323	5	5n+	5n+	NUM
ejpam-5974	323	6	3	3	NUM
ejpam-5974	323	7	.	.	PUNCT
ejpam-5974	324	1	acknowledgements	acknowledgement	NOUN
ejpam-5974	324	2	the	the	DET
ejpam-5974	324	3	authors	author	NOUN
ejpam-5974	324	4	would	would	AUX
ejpam-5974	324	5	like	like	VERB
ejpam-5974	324	6	to	to	PART
ejpam-5974	324	7	extend	extend	VERB
ejpam-5974	324	8	their	their	PRON
ejpam-5974	324	9	heartfelt	heartfelt	ADJ
ejpam-5974	324	10	gratitude	gratitude	NOUN
ejpam-5974	324	11	and	and	CCONJ
ejpam-5974	324	12	appreciation	appreciation	NOUN
ejpam-5974	324	13	to	to	ADP
ejpam-5974	324	14	the	the	DET
ejpam-5974	324	15	people	people	NOUN
ejpam-5974	324	16	who	who	PRON
ejpam-5974	324	17	are	be	AUX
ejpam-5974	324	18	beyond	beyond	ADP
ejpam-5974	324	19	the	the	DET
ejpam-5974	324	20	success	success	NOUN
ejpam-5974	324	21	of	of	ADP
ejpam-5974	324	22	this	this	DET
ejpam-5974	324	23	paper	paper	NOUN
ejpam-5974	324	24	.	.	PUNCT
ejpam-5974	325	1	also	also	ADV
ejpam-5974	325	2	,	,	PUNCT
ejpam-5974	325	3	the	the	DET
ejpam-5974	325	4	authors	author	NOUN
ejpam-5974	325	5	would	would	AUX
ejpam-5974	325	6	like	like	VERB
ejpam-5974	325	7	to	to	PART
ejpam-5974	325	8	express	express	VERB
ejpam-5974	325	9	their	their	PRON
ejpam-5974	325	10	profound	profound	ADJ
ejpam-5974	325	11	gratitude	gratitude	NOUN
ejpam-5974	325	12	to	to	ADP
ejpam-5974	325	13	the	the	DET
ejpam-5974	325	14	department	department	NOUN
ejpam-5974	325	15	of	of	ADP
ejpam-5974	325	16	science	science	NOUN
ejpam-5974	325	17	and	and	CCONJ
ejpam-5974	325	18	technology	technology	NOUN
ejpam-5974	325	19	-	-	PUNCT
ejpam-5974	325	20	science	science	NOUN
ejpam-5974	325	21	education	education	PROPN
ejpam-5974	325	22	institute	institute	PROPN
ejpam-5974	325	23	science	science	PROPN
ejpam-5974	325	24	and	and	CCONJ
ejpam-5974	325	25	technology	technology	NOUN
ejpam-5974	325	26	regional	regional	ADJ
ejpam-5974	325	27	alliance	alliance	NOUN
ejpam-5974	325	28	of	of	ADP
ejpam-5974	325	29	universities	university	NOUN
ejpam-5974	325	30	for	for	ADP
ejpam-5974	325	31	inclusive	inclusive	ADJ
ejpam-5974	325	32	national	national	ADJ
ejpam-5974	325	33	development	development	NOUN
ejpam-5974	325	34	(	(	PUNCT
ejpam-5974	325	35	dost	dost	NOUN
ejpam-5974	325	36	-	-	PUNCT
ejpam-5974	325	37	sei	sei	ADJ
ejpam-5974	325	38	strand	strand	NOUN
ejpam-5974	325	39	)	)	PUNCT
ejpam-5974	325	40	for	for	ADP
ejpam-5974	325	41	their	their	PRON
ejpam-5974	325	42	invaluable	invaluable	ADJ
ejpam-5974	325	43	assistance	assistance	NOUN
ejpam-5974	325	44	throughout	throughout	ADP
ejpam-5974	325	45	the	the	DET
ejpam-5974	325	46	research	research	NOUN
ejpam-5974	325	47	process	process	NOUN
ejpam-5974	325	48	.	.	PUNCT
ejpam-5974	326	1	references	reference	NOUN
ejpam-5974	326	2	[	[	X
ejpam-5974	326	3	1	1	NUM
ejpam-5974	326	4	]	]	PUNCT
ejpam-5974	326	5	m.	m.	PROPN
ejpam-5974	326	6	e.	e.	PROPN
ejpam-5974	326	7	j.	j.	PROPN
ejpam-5974	326	8	newman	newman	PROPN
ejpam-5974	326	9	.	.	PUNCT
ejpam-5974	327	1	the	the	DET
ejpam-5974	327	2	structure	structure	NOUN
ejpam-5974	327	3	and	and	CCONJ
ejpam-5974	327	4	function	function	NOUN
ejpam-5974	327	5	of	of	ADP
ejpam-5974	327	6	complex	complex	ADJ
ejpam-5974	327	7	networks	network	NOUN
ejpam-5974	327	8	.	.	PUNCT
ejpam-5974	328	1	siam	siam	PROPN
ejpam-5974	328	2	review	review	PROPN
ejpam-5974	328	3	,	,	PUNCT
ejpam-5974	328	4	45(2):167–256	45(2):167–256	PROPN
ejpam-5974	328	5	,	,	PUNCT
ejpam-5974	328	6	2003	2003	NUM
ejpam-5974	328	7	.	.	PUNCT
ejpam-5974	329	1	[	[	X
ejpam-5974	329	2	2	2	NUM
ejpam-5974	329	3	]	]	X
ejpam-5974	329	4	r.	r.	NOUN
ejpam-5974	329	5	sango	sango	PROPN
ejpam-5974	329	6	and	and	CCONJ
ejpam-5974	329	7	i.	i.	PROPN
ejpam-5974	329	8	cabahug	cabahug	PROPN
ejpam-5974	329	9	jr	jr	PROPN
ejpam-5974	329	10	.	.	PROPN
ejpam-5974	329	11	on	on	ADP
ejpam-5974	329	12	the	the	DET
ejpam-5974	329	13	density	density	NOUN
ejpam-5974	329	14	of	of	ADP
ejpam-5974	329	15	some	some	DET
ejpam-5974	329	16	graphs	graph	NOUN
ejpam-5974	329	17	and	and	CCONJ
ejpam-5974	329	18	corona	corona	NOUN
ejpam-5974	329	19	graphs	graph	NOUN
ejpam-5974	329	20	.	.	PUNCT
ejpam-5974	330	1	asian	asian	ADJ
ejpam-5974	330	2	research	research	PROPN
ejpam-5974	330	3	journal	journal	NOUN
ejpam-5974	330	4	of	of	ADP
ejpam-5974	330	5	mathematics	mathematic	NOUN
ejpam-5974	330	6	,	,	PUNCT
ejpam-5974	330	7	21(6):1–6	21(6):1–6	NOUN
ejpam-5974	330	8	,	,	PUNCT
ejpam-5974	330	9	2025	2025	NUM
ejpam-5974	330	10	.	.	PUNCT
ejpam-5974	331	1	[	[	X
ejpam-5974	331	2	3	3	NUM
ejpam-5974	331	3	]	]	X
ejpam-5974	331	4	r.	r.	PROPN
ejpam-5974	331	5	eballe	eballe	PROPN
ejpam-5974	331	6	and	and	CCONJ
ejpam-5974	331	7	i.	i.	PROPN
ejpam-5974	331	8	cabahug	cabahug	PROPN
ejpam-5974	331	9	jr	jr	PROPN
ejpam-5974	331	10	.	.	PROPN
ejpam-5974	331	11	closeness	closeness	NOUN
ejpam-5974	331	12	centrality	centrality	NOUN
ejpam-5974	331	13	of	of	ADP
ejpam-5974	331	14	some	some	DET
ejpam-5974	331	15	graph	graph	NOUN
ejpam-5974	331	16	families	family	NOUN
ejpam-5974	331	17	.	.	PUNCT
ejpam-5974	332	1	international	international	ADJ
ejpam-5974	332	2	journal	journal	PROPN
ejpam-5974	332	3	of	of	ADP
ejpam-5974	332	4	mathematics	mathematics	PROPN
ejpam-5974	332	5	and	and	CCONJ
ejpam-5974	332	6	statistics	statistic	NOUN
ejpam-5974	332	7	invention	invention	NOUN
ejpam-5974	332	8	(	(	PUNCT
ejpam-5974	332	9	ijmsi	ijmsi	NOUN
ejpam-5974	332	10	)	)	PUNCT
ejpam-5974	332	11	,	,	PUNCT
ejpam-5974	332	12	16(4):127–134	16(4):127–134	NUM
ejpam-5974	332	13	,	,	PUNCT
ejpam-5974	332	14	2021	2021	NUM
ejpam-5974	332	15	.	.	PUNCT
ejpam-5974	333	1	[	[	X
ejpam-5974	333	2	4	4	NUM
ejpam-5974	333	3	]	]	X
ejpam-5974	333	4	r.	r.	PROPN
ejpam-5974	333	5	eballe	eballe	PROPN
ejpam-5974	333	6	,	,	PUNCT
ejpam-5974	333	7	c.	c.	PROPN
ejpam-5974	333	8	m.	m.	PROPN
ejpam-5974	333	9	balingit	balingit	PROPN
ejpam-5974	333	10	,	,	PUNCT
ejpam-5974	333	11	i.	i.	PROPN
ejpam-5974	333	12	cabahug	cabahug	PROPN
ejpam-5974	333	13	jr	jr	PROPN
ejpam-5974	333	14	.	.	PROPN
ejpam-5974	333	15	,	,	PUNCT
ejpam-5974	333	16	a.	a.	PROPN
ejpam-5974	333	17	l.	l.	PROPN
ejpam-5974	333	18	flores	flores	PROPN
ejpam-5974	333	19	,	,	PUNCT
ejpam-5974	333	20	s.	s.	PROPN
ejpam-5974	333	21	m.	m.	PROPN
ejpam-5974	333	22	lumpayao	lumpayao	PROPN
ejpam-5974	333	23	,	,	PUNCT
ejpam-5974	333	24	b.	b.	PROPN
ejpam-5974	333	25	penalosa	penalosa	PROPN
ejpam-5974	333	26	,	,	PUNCT
ejpam-5974	333	27	g.	g.	PROPN
ejpam-5974	333	28	a.	a.	PROPN
ejpam-5974	333	29	tampipi	tampipi	PROPN
ejpam-5974	333	30	,	,	PUNCT
ejpam-5974	333	31	and	and	CCONJ
ejpam-5974	333	32	c.	c.	PROPN
ejpam-5974	333	33	villarta	villarta	PROPN
ejpam-5974	333	34	.	.	PUNCT
ejpam-5974	334	1	closeness	closeness	NOUN
ejpam-5974	334	2	centrality	centrality	NOUN
ejpam-5974	334	3	in	in	ADP
ejpam-5974	334	4	graph	graph	NOUN
ejpam-5974	334	5	products	product	NOUN
ejpam-5974	334	6	.	.	PUNCT
ejpam-5974	335	1	advances	advance	NOUN
ejpam-5974	335	2	and	and	CCONJ
ejpam-5974	335	3	applications	application	NOUN
ejpam-5974	335	4	in	in	ADP
ejpam-5974	335	5	discrete	discrete	ADJ
ejpam-5974	335	6	mathematics	mathematic	NOUN
ejpam-5974	335	7	,	,	PUNCT
ejpam-5974	335	8	39(1):29–41	39(1):29–41	NUM
ejpam-5974	335	9	,	,	PUNCT
ejpam-5974	335	10	2023	2023	NUM
ejpam-5974	335	11	.	.	PUNCT
ejpam-5974	336	1	[	[	X
ejpam-5974	336	2	5	5	NUM
ejpam-5974	336	3	]	]	PUNCT
ejpam-5974	336	4	l.	l.	PROPN
ejpam-5974	336	5	toladro	toladro	PROPN
ejpam-5974	336	6	and	and	CCONJ
ejpam-5974	336	7	i.	i.	PROPN
ejpam-5974	336	8	cabahug	cabahug	PROPN
ejpam-5974	336	9	jr	jr	PROPN
ejpam-5974	336	10	.	.	PUNCT
ejpam-5974	336	11	proximity	proximity	NOUN
ejpam-5974	336	12	prestige	prestige	NOUN
ejpam-5974	336	13	of	of	ADP
ejpam-5974	336	14	a	a	DET
ejpam-5974	336	15	vertex	vertex	NOUN
ejpam-5974	336	16	in	in	ADP
ejpam-5974	336	17	some	some	DET
ejpam-5974	336	18	graph	graph	NOUN
ejpam-5974	336	19	families	family	NOUN
ejpam-5974	336	20	.	.	PUNCT
ejpam-5974	337	1	european	european	ADJ
ejpam-5974	337	2	journal	journal	PROPN
ejpam-5974	337	3	of	of	ADP
ejpam-5974	337	4	pure	pure	ADJ
ejpam-5974	337	5	and	and	CCONJ
ejpam-5974	337	6	applied	applied	ADJ
ejpam-5974	337	7	mathematics	mathematic	NOUN
ejpam-5974	337	8	,	,	PUNCT
ejpam-5974	337	9	18(2):5943	18(2):5943	NUM
ejpam-5974	337	10	,	,	PUNCT
ejpam-5974	337	11	2025	2025	NUM
ejpam-5974	337	12	.	.	PUNCT
ejpam-5974	338	1	[	[	X
ejpam-5974	338	2	6	6	NUM
ejpam-5974	338	3	]	]	PUNCT
ejpam-5974	338	4	g.	g.	PROPN
ejpam-5974	338	5	chartrand	chartrand	PROPN
ejpam-5974	338	6	,	,	PUNCT
ejpam-5974	338	7	l.	l.	PROPN
ejpam-5974	338	8	m.	m.	PROPN
ejpam-5974	338	9	lesniak	lesniak	PROPN
ejpam-5974	338	10	,	,	PUNCT
ejpam-5974	338	11	and	and	CCONJ
ejpam-5974	338	12	p.	p.	PROPN
ejpam-5974	338	13	zhang	zhang	PROPN
ejpam-5974	338	14	.	.	PUNCT
ejpam-5974	339	1	graphs	graph	NOUN
ejpam-5974	339	2	and	and	CCONJ
ejpam-5974	339	3	digraphs	digraph	NOUN
ejpam-5974	339	4	.	.	PUNCT
ejpam-5974	340	1	crc	crc	PROPN
ejpam-5974	340	2	press	press	PROPN
ejpam-5974	340	3	,	,	PUNCT
ejpam-5974	340	4	new	new	PROPN
ejpam-5974	340	5	york	york	PROPN
ejpam-5974	340	6	,	,	PUNCT
ejpam-5974	340	7	6	6	NUM
ejpam-5974	340	8	edition	edition	NOUN
ejpam-5974	340	9	,	,	PUNCT
ejpam-5974	340	10	2015	2015	NUM
ejpam-5974	340	11	.	.	PUNCT
ejpam-5974	341	1	[	[	X
ejpam-5974	341	2	7	7	X
ejpam-5974	341	3	]	]	X
ejpam-5974	341	4	f.	f.	PROPN
ejpam-5974	341	5	harrary	harrary	PROPN
ejpam-5974	341	6	.	.	PUNCT
ejpam-5974	342	1	graph	graph	NOUN
ejpam-5974	342	2	theory	theory	NOUN
ejpam-5974	342	3	.	.	PUNCT
ejpam-5974	343	1	addison	addison	PROPN
ejpam-5974	343	2	-	-	PUNCT
ejpam-5974	343	3	wesley	wesley	PROPN
ejpam-5974	343	4	publishing	publishing	PROPN
ejpam-5974	343	5	company	company	PROPN
ejpam-5974	343	6	,	,	PUNCT
ejpam-5974	343	7	inc	inc	PROPN
ejpam-5974	343	8	.	.	PROPN
ejpam-5974	343	9	,	,	PUNCT
ejpam-5974	343	10	united	united	PROPN
ejpam-5974	343	11	states	states	PROPN
ejpam-5974	343	12	of	of	ADP
ejpam-5974	343	13	america	america	PROPN
ejpam-5974	343	14	,	,	PUNCT
ejpam-5974	343	15	1969	1969	NUM
ejpam-5974	343	16	.	.	PUNCT
ejpam-5974	344	1	[	[	X
ejpam-5974	344	2	8	8	NUM
ejpam-5974	344	3	]	]	X
ejpam-5974	344	4	h.	h.	PROPN
ejpam-5974	344	5	komarullah	komarullah	PROPN
ejpam-5974	344	6	,	,	PUNCT
ejpam-5974	344	7	j.	j.	PROPN
ejpam-5974	344	8	halilim	halilim	PROPN
ejpam-5974	344	9	,	,	PUNCT
ejpam-5974	344	10	and	and	CCONJ
ejpam-5974	344	11	k.	k.	PROPN
ejpam-5974	344	12	santoso	santoso	PROPN
ejpam-5974	344	13	.	.	PUNCT
ejpam-5974	345	1	on	on	ADP
ejpam-5974	345	2	the	the	DET
ejpam-5974	345	3	minimum	minimum	ADJ
ejpam-5974	345	4	span	span	NOUN
ejpam-5974	345	5	of	of	ADP
ejpam-5974	345	6	cone	cone	NOUN
ejpam-5974	345	7	,	,	PUNCT
ejpam-5974	345	8	tadpole	tadpole	NOUN
ejpam-5974	345	9	,	,	PUNCT
ejpam-5974	345	10	and	and	CCONJ
ejpam-5974	345	11	barbell	barbell	NOUN
ejpam-5974	345	12	graphs	graph	NOUN
ejpam-5974	345	13	.	.	PUNCT
ejpam-5974	346	1	in	in	ADP
ejpam-5974	346	2	proceedings	proceeding	NOUN
ejpam-5974	346	3	of	of	ADP
ejpam-5974	346	4	international	international	ADJ
ejpam-5974	346	5	conference	conference	NOUN
ejpam-5974	346	6	on	on	ADP
ejpam-5974	346	7	mathematics	mathematic	NOUN
ejpam-5974	346	8	,	,	PUNCT
ejpam-5974	346	9	geometry	geometry	NOUN
ejpam-5974	346	10	,	,	PUNCT
ejpam-5974	346	11	statistics	statistic	NOUN
ejpam-5974	346	12	and	and	CCONJ
ejpam-5974	346	13	computation	computation	NOUN
ejpam-5974	346	14	,	,	PUNCT
ejpam-5974	346	15	2021	2021	NUM
ejpam-5974	346	16	.	.	PUNCT
ejpam-5974	347	1	[	[	X
ejpam-5974	347	2	9	9	NUM
ejpam-5974	347	3	]	]	PUNCT
ejpam-5974	347	4	j.	j.	PROPN
ejpam-5974	347	5	gallian	gallian	PROPN
ejpam-5974	347	6	.	.	PUNCT
ejpam-5974	348	1	dynamic	dynamic	ADJ
ejpam-5974	348	2	survey	survey	NOUN
ejpam-5974	348	3	of	of	ADP
ejpam-5974	348	4	graph	graph	NOUN
ejpam-5974	348	5	labeling	labeling	NOUN
ejpam-5974	348	6	.	.	PUNCT
ejpam-5974	349	1	electronic	electronic	ADJ
ejpam-5974	349	2	journal	journal	NOUN
ejpam-5974	349	3	of	of	ADP
ejpam-5974	349	4	combinatorics	combinatoric	NOUN
ejpam-5974	349	5	,	,	PUNCT
ejpam-5974	349	6	19(4):337–348	19(4):337–348	PROPN
ejpam-5974	349	7	,	,	PUNCT
ejpam-5974	349	8	2000	2000	NUM
ejpam-5974	349	9	.	.	PUNCT
ejpam-5974	350	1	[	[	X
ejpam-5974	350	2	10	10	NUM
ejpam-5974	350	3	]	]	PUNCT
ejpam-5974	350	4	k.	k.	PROPN
ejpam-5974	350	5	vaithilingan	vaithilingan	PROPN
ejpam-5974	350	6	.	.	PUNCT
ejpam-5974	351	1	difference	difference	NOUN
ejpam-5974	351	2	labeling	labeling	NOUN
ejpam-5974	351	3	of	of	ADP
ejpam-5974	351	4	some	some	DET
ejpam-5974	351	5	graph	graph	NOUN
ejpam-5974	351	6	families	family	NOUN
ejpam-5974	351	7	.	.	PUNCT
ejpam-5974	352	1	international	international	ADJ
ejpam-5974	352	2	journal	journal	PROPN
ejpam-5974	352	3	of	of	ADP
ejpam-5974	352	4	mathematics	mathematics	PROPN
ejpam-5974	352	5	and	and	CCONJ
ejpam-5974	352	6	statistics	statistic	NOUN
ejpam-5974	352	7	invention	invention	NOUN
ejpam-5974	352	8	(	(	PUNCT
ejpam-5974	352	9	ijmsi	ijmsi	NOUN
ejpam-5974	352	10	)	)	PUNCT
ejpam-5974	352	11	,	,	PUNCT
ejpam-5974	352	12	2014	2014	NUM
ejpam-5974	352	13	.	.	PUNCT
ejpam-5974	353	1	r.	r.	PROPN
ejpam-5974	353	2	sango	sango	PROPN
ejpam-5974	353	3	,	,	PUNCT
ejpam-5974	353	4	i.	i.	NOUN
ejpam-5974	353	5	cabahug	cabahug	PROPN
ejpam-5974	353	6	,	,	PUNCT
ejpam-5974	353	7	jr	jr	PROPN
ejpam-5974	353	8	/	/	SYM
ejpam-5974	353	9	eur	eur	PROPN
ejpam-5974	353	10	.	.	PUNCT
ejpam-5974	354	1	j.	j.	PROPN
ejpam-5974	354	2	pure	pure	PROPN
ejpam-5974	354	3	appl	appl	PROPN
ejpam-5974	354	4	.	.	PROPN
ejpam-5974	354	5	math	math	PROPN
ejpam-5974	354	6	,	,	PUNCT
ejpam-5974	354	7	18	18	NUM
ejpam-5974	354	8	(	(	PUNCT
ejpam-5974	354	9	3	3	NUM
ejpam-5974	354	10	)	)	PUNCT
ejpam-5974	354	11	(	(	PUNCT
ejpam-5974	354	12	2025	2025	NUM
ejpam-5974	354	13	)	)	PUNCT
ejpam-5974	354	14	,	,	PUNCT
ejpam-5974	354	15	5974	5974	NUM
ejpam-5974	354	16	16	16	NUM
ejpam-5974	354	17	of	of	ADP
ejpam-5974	354	18	16	16	NUM
ejpam-5974	355	1	[	[	X
ejpam-5974	355	2	11	11	NUM
ejpam-5974	355	3	]	]	X
ejpam-5974	355	4	r.	r.	PROPN
ejpam-5974	355	5	frucht	frucht	PROPN
ejpam-5974	355	6	.	.	PUNCT
ejpam-5974	356	1	graceful	graceful	ADJ
ejpam-5974	356	2	numbering	numbering	NOUN
ejpam-5974	356	3	of	of	ADP
ejpam-5974	356	4	wheels	wheel	NOUN
ejpam-5974	356	5	and	and	CCONJ
ejpam-5974	356	6	related	related	ADJ
ejpam-5974	356	7	graphs	graph	NOUN
ejpam-5974	356	8	.	.	PUNCT
ejpam-5974	357	1	annals	annal	NOUN
ejpam-5974	357	2	of	of	ADP
ejpam-5974	357	3	the	the	DET
ejpam-5974	357	4	new	new	PROPN
ejpam-5974	357	5	york	york	PROPN
ejpam-5974	357	6	academy	academy	PROPN
ejpam-5974	357	7	of	of	ADP
ejpam-5974	357	8	sciences	sciences	PROPN
ejpam-5974	357	9	,	,	PUNCT
ejpam-5974	357	10	319(1):219–229	319(1):219–229	NUM
ejpam-5974	357	11	,	,	PUNCT
ejpam-5974	357	12	1979	1979	NUM
ejpam-5974	357	13	.	.	PUNCT
ejpam-5974	358	1	[	[	X
ejpam-5974	358	2	12	12	NUM
ejpam-5974	358	3	]	]	PUNCT
ejpam-5974	358	4	w.	w.	PROPN
ejpam-5974	358	5	c.	c.	PROPN
ejpam-5974	358	6	chen	chen	PROPN
ejpam-5974	358	7	,	,	PUNCT
ejpam-5974	358	8	h.	h.	PROPN
ejpam-5974	358	9	i.	i.	PROPN
ejpam-5974	358	10	hu	hu	PROPN
ejpam-5974	358	11	,	,	PUNCT
ejpam-5974	358	12	and	and	CCONJ
ejpam-5974	358	13	y.	y.	PROPN
ejpam-5974	358	14	n.	n.	PROPN
ejpam-5974	358	15	yeh	yeh	PROPN
ejpam-5974	358	16	.	.	PUNCT
ejpam-5974	359	1	operations	operation	NOUN
ejpam-5974	359	2	of	of	ADP
ejpam-5974	359	3	interlaced	interlaced	ADJ
ejpam-5974	359	4	trees	tree	NOUN
ejpam-5974	359	5	and	and	CCONJ
ejpam-5974	359	6	graceful	graceful	ADJ
ejpam-5974	359	7	trees	tree	NOUN
ejpam-5974	359	8	.	.	PUNCT
ejpam-5974	360	1	southeast	southeast	ADJ
ejpam-5974	360	2	asian	asian	ADJ
ejpam-5974	360	3	bulletin	bulletin	NOUN
ejpam-5974	360	4	of	of	ADP
ejpam-5974	360	5	mathematics	mathematic	NOUN
ejpam-5974	360	6	,	,	PUNCT
ejpam-5974	360	7	21(4):337–348	21(4):337–348	NUM
ejpam-5974	360	8	,	,	PUNCT
ejpam-5974	360	9	1997	1997	NUM
ejpam-5974	360	10	.	.	PUNCT
ejpam-5974	361	1	[	[	X
ejpam-5974	361	2	13	13	NUM
ejpam-5974	361	3	]	]	PUNCT
ejpam-5974	361	4	m.	m.	NOUN
ejpam-5974	361	5	dinorog	dinorog	NOUN
ejpam-5974	361	6	and	and	CCONJ
ejpam-5974	361	7	i.	i.	PROPN
ejpam-5974	361	8	cabahug	cabahug	PROPN
ejpam-5974	361	9	.	.	PUNCT
ejpam-5974	362	1	rings	ring	NOUN
ejpam-5974	362	2	and	and	CCONJ
ejpam-5974	362	3	domination	domination	NOUN
ejpam-5974	362	4	number	number	NOUN
ejpam-5974	362	5	of	of	ADP
ejpam-5974	362	6	some	some	DET
ejpam-5974	362	7	myscielski	myscielski	ADJ
ejpam-5974	362	8	graphs	graph	NOUN
ejpam-5974	362	9	.	.	PUNCT
ejpam-5974	363	1	asian	asian	ADJ
ejpam-5974	363	2	research	research	PROPN
ejpam-5974	363	3	journal	journal	NOUN
ejpam-5974	363	4	of	of	ADP
ejpam-5974	363	5	mathematics	mathematic	NOUN
ejpam-5974	363	6	,	,	PUNCT
ejpam-5974	363	7	18(12):16–26	18(12):16–26	NUM
ejpam-5974	363	8	,	,	PUNCT
ejpam-5974	363	9	2022	2022	NUM
ejpam-5974	363	10	.	.	PUNCT
ejpam-5974	364	1	[	[	X
ejpam-5974	364	2	14	14	NUM
ejpam-5974	364	3	]	]	X
ejpam-5974	364	4	r.	r.	PROPN
ejpam-5974	364	5	diestel	diestel	PROPN
ejpam-5974	364	6	.	.	PUNCT
ejpam-5974	365	1	graph	graph	NOUN
ejpam-5974	365	2	theory	theory	NOUN
ejpam-5974	365	3	:	:	PUNCT
ejpam-5974	365	4	graduate	graduate	NOUN
ejpam-5974	365	5	text	text	NOUN
ejpam-5974	365	6	in	in	ADP
ejpam-5974	365	7	mathematics	mathematics	PROPN
ejpam-5974	365	8	.	.	PUNCT
ejpam-5974	366	1	springer	springer	NOUN
ejpam-5974	366	2	,	,	PUNCT
ejpam-5974	366	3	5	5	NUM
ejpam-5974	366	4	edition	edition	NOUN
ejpam-5974	366	5	,	,	PUNCT
ejpam-5974	366	6	2017	2017	NUM
ejpam-5974	366	7	.	.	PUNCT